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A fundamental domain of the ring of integers of the field K obtained from Q by adjoining a root of x³ − x² − 2x + 1. This fundamental domain sits inside K ⊗_QR. The discriminant of K is 49 = 7². Accordingly, the volume of the fundamental domain is 7 and K is only ramified at 7.

In mathematics, the discriminant of an algebraic number field is a numerical invariant that, loosely speaking, measures the size of the (ring of integers of the) algebraic number field. More specifically, it is related to the volume of the fundamental domain of the ring of integers, and it regulates which primes are ramified.

The discriminant is one of the most basic invariants of a number field, and occurs in several important analytic formulas such as the functional equation of the Dedekind zeta function of K, and the analytic class number formula for K. An old theorem of Hermite states that there are only finitely many number fields of bounded discriminant, however determining this quantity is still an open problem, and the subject of current research.^[1]

The discriminant of K can be referred to as the absolute discriminant of K to distinguish it from the relative discriminant of an extension K/L of number fields. The latter is an ideal in the ring of integers of L, and like the absolute discriminant it indicates which primes are ramified in K/L. It is a generalization of the absolute discriminant allowing for L to be bigger than Q; in fact, when L = Q, the relative discriminant of K/Q is the principal ideal of Z generated by the absolute discriminant of K.

Definition

Let K be an algebraic number field, and let O_K be its ring of integers. Let b₁, ..., b_n be an integral basis of O_K (i.e. a basis as a Z-module), and let {σ₁, ..., σ_n} be the set of embeddings of K into the complex numbers (i.e. injective ring homomorphisms K → C). The discriminant of K is the square of the determinant of the n by n matrix B whose (i,j)-entry is σ_i(b_j). Symbolically,

Δ_{K} = {(\det (\begin{array}{cccc} σ_{1} (b_{1}) & σ_{1} (b_{2}) & \dots & σ_{1} (b_{n}) \\ σ_{2} (b_{1}) & ⋱ & ⋮ \\ ⋮ & ⋱ & ⋮ \\ σ_{n} (b_{1}) & \dots & \dots & σ_{n} (b_{n}) \end{array}))}^{2} .

Equivalently, the trace from K to Q can be used. Specifically, define the trace form to be the matrix whose (i,j)-entry is Tr_K/Q(b_ib_j). This matrix equals B^TB, so the discriminant of K is the determinant of this matrix.

Examples

Quadratic number fields: let d be a square-free integer, then the discriminant of $K = Q (\sqrt{d})$ is^[2]

Δ_{K} = {\begin{array}{ll} d & if d \equiv 1 (\mod 4) \\ 4 d & if d \equiv 2, 3 (\mod 4) . \end{array}

An integer that occurs as the discriminant of a quadratic number field is called a fundamental discriminant.^[3]

Cyclotomic fields: let n > 2 be an integer, let ζ_n be a primitive nth root of unity, and let K_n = Q(ζ_n) be the nth cyclotomic field. The discriminant of K_n is given by^[4]^[2]

Δ_{K_{n}} = (- 1)^{φ (n) / 2} \frac{n^{φ (n)}}{\prod_{p | n} p^{φ (n) / (p - 1)}}

where

φ (n)

is Euler's totient function, and the product in the denominator is over primes p dividing n.

Power bases: In the case where the ring of integers has a power integral basis, that is, can be written as O_K = Z[α], the discriminant of K is equal to the discriminant of the minimal polynomial of α. To see this, one can chose the integral basis of O_K to be b₁ = 1, b₂ = α, b₃ = α², ..., b_n = αⁿ⁻¹. Then, the matrix in the definition is the Vandermonde matrix associated to α_i = σ_i(α), whose determinant squared is

\prod_{1 \leq i < j \leq n} (α_{i} - α_{j})^{2}

which is exactly the definition of the discriminant of the minimal polynomial.

Let K = Q(α) be the number field obtained by adjoining a root α of the polynomial x³ − x² − 2x − 8. This is Richard Dedekind's original example of a number field whose ring of integers does not possess a power basis. An integral basis is given by {1, α, α(α + 1)/2} and the discriminant of K is −503.^[5]^[6]
Repeated discriminants: the discriminant of a quadratic field uniquely identifies it, but this is not true, in general, for higher-degree number fields. For example, there are two non-isomorphic cubic fields of discriminant 3969. They are obtained by adjoining a root of the polynomial Analyst Programmer Alfonzo Crosser from Newcastle, usually spends time with interests which include frisbee golf - frolf, property developers properties for sale in singapore singapore and collecting music albums. Ended up in recent past visiting Puerto-Princesa Subterranean River National Park. or Analyst Programmer Alfonzo Crosser from Newcastle, usually spends time with interests which include frisbee golf - frolf, property developers properties for sale in singapore singapore and collecting music albums. Ended up in recent past visiting Puerto-Princesa Subterranean River National Park., respectively.^[7]

Basic results

Brill's theorem:^[8] The sign of the discriminant is (−1)^r₂ where r₂ is the number of complex places of K.^[9]
A prime p ramifies in K if, and only if, p divides Δ_K .^[10]
Stickelberger's theorem:^[11]

Δ_{K} \equiv 0 or 1 (\mod 4) .

Minkowski's bound:^[12] Let n denote the degree of the extension K/Q and r₂ the number of complex places of K, then

| Δ_{K} |^{1 / 2} \geq \frac{n^{n}}{n!} {(\frac{π}{4})}^{r_{2}} \geq \frac{n^{n}}{n!} {(\frac{π}{4})}^{n / 2} .

Minkowski's theorem:^[13] If K is not Q, then |Δ_K| > 1 (this follows directly from the Minkowski bound).
Hermite–Minkowski theorem:^[14] Let N be a positive integer. There are only finitely many (up to isomorphisms) algebraic number fields K with |Δ_K| < N. Again, this follows from the Minkowski bound together Hermite's theorem (There are only finitely many algebraic number fields with prescribed discriminant).

History

Richard Dedekind showed that every number field possesses an integral basis, allowing him to define the discriminant of an arbitrary number field.^[15]

The definition of the discriminant of a general algebraic number field, K, was given by Dedekind in 1871.^[15] At this point, he already knew the relationship between the discriminant and ramification.^[16]

Hermite's theorem predates the general definition of the discriminant with Charles Hermite publishing a proof of it in 1857.Template:Sfn In 1877, Alexander von Brill determined the sign of the discriminant.Template:Sfn Leopold Kronecker first stated Minkowski's theorem in 1882,Template:Sfn though the first proof was given by Hermann Minkowski in 1891.Template:Sfn In the same year, Minkowski published his bound on the discriminant.Template:Sfn Near the end of the nineteenth century, Ludwig Stickelberger obtained his theorem on the residue of the discriminant modulo four.Template:Sfn^[17]

Relative discriminant

The discriminant defined above is sometimes referred to as the absolute discriminant of K to distinguish it from the relative discriminant Δ_K/L of an extension of number fields K/L, which is an ideal in O_L. The relative discriminant is defined in a fashion similar to the absolute discriminant, but must take into account that ideals in O_L may not be principal and that there may not be an integral basis of K/L. Let {σ₁, ..., σ_n} be the set of embeddings of K into C which are the identity on L. If b₁, ..., b_n is any basis of K over L, let d(b₁, ..., b_n) be the square of the determinant of the n by n matrix whose (i,j)-entry is σ_i(b_j). Then, the relative discriminant of K/L is the ideal generated by the d(b₁, ..., b_n) as {b₁, ..., b_n} varies over all bases of K/L with the property that b_i ∈ O_K for all i. Alternatively, the relative discriminant of K/L is the norm of the different of K/L.^[18] When L = Q, the relative discriminant Δ_K/Q is the principal ideal of Z generated by the absolute discriminant Δ_K . In a tower of fields K/L/F the relative discriminants are related by

Δ_{K / F} = 𝒩_{L / F} (Δ_{K / L}) Δ_{L / F}^{[K : L]}

where $𝒩$ denotes relative norm.^[19]

Ramification

The relative discriminant regulates the ramification data of the field extension K/L. A prime ideal p of L ramifies in K if, and only if, it divides the relative discriminant Δ_K/L. An extension is unramified if, and only if, the discriminant is the unit ideal.^[18] The Minkowski bound above shows that there are no non-trivial unramified extensions of Q. Fields larger than Q may have unramified extensions, for example, for any field with class number greater than one, its Hilbert class field is a non-trivial unramified extension.

Root discriminant

The root discriminant of a number field, K, of degree n, often denoted rd_K, is defined as the n-th root of the absolute value of the (absolute) discriminant of K.^[20] The relation between relative discriminants in a tower of fields shows that the root discriminant does not change in an unramified extension. The existence of a class field tower provides bounds on the root discriminant: the existence of an infinite class field tower over Q(√-m) where m = 3·5·7·11·19 shows that there are infinitely many fields with root discriminant 2√m ≈ 296.276.^[21] If we let r and 2s be the number of real and complex embeddings, so that n = r + 2s, put ρ = r/n and σ = 2s/n. Set α(ρ, σ) to be the infimum of rd_K for K with (r', 2s') = (ρn, σn). We have^[21]

α (ρ, σ) \geq 60 . 8^{ρ} 22 . 3^{σ}

and on the assumption of the generalized Riemann hypothesis

α (ρ, σ) \geq 215 . 3^{ρ} 44 . 7^{σ} .

So we have α(0,1) < 296.276. Martinet has shown α(0,1) < 93 and α(1,0) < 1059.^[21]^[22] Template:Harvnb proves that for totally real fields, the root discriminant is > 14, with 1229 exceptions.

Relation to other quantities

When embedded into $K \otimes_{Q} R$ , the volume of the fundamental domain of O_K is $\sqrt{| Δ_{K} |}$ (sometimes a different measure is used and the volume obtained is $2^{- r_{2}} \sqrt{| Δ_{K} |}$ , where r₂ is the number of complex places of K).
Due to its appearance in this volume, the discriminant also appears in the functional equation of the Dedekind zeta function of K, and hence in the analytic class number formula, and the Brauer–Siegel theorem.
The relative discriminant of K/L is the Artin conductor of the regular representation of the Galois group of K/L. This provides a relation to the Artin conductors of the characters of the Galois group of K/L, called the conductor-discriminant formula.^[23]

Notes

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References

Primary sources

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Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

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Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010

Secondary sources

Template:Bourbaki EHM
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Template:Neukirch ANT
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

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15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010

Direct image functor: Difference between revisions

Latest revision as of 16:29, 11 November 2013

Contents

Definition

Examples

Basic results

History

Relative discriminant

Ramification

Root discriminant

Relation to other quantities

Notes

References

Primary sources

Secondary sources

Further reading

Navigation menu

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+{{Redirect|Brill's theorem|the result in algebraic geometry|Brill–Noether theorem}}
+[[File:Discriminant49CubicFieldFundamentalDomain.png|thumb|300px|right|A fundamental domain of the ring of integers of the field ''K'' obtained from '''Q''' by adjoining a root of ''x''<sup>3</sup>&nbsp;&minus;&nbsp;''x''<sup>2</sup>&nbsp;&minus;&nbsp;2''x''&nbsp;+&nbsp;1. This fundamental domain sits inside ''K''&nbsp;&otimes;<sub>'''Q'''</sub>'''R'''. The discriminant of ''K'' is 49&nbsp;=&nbsp;7<sup>2</sup>. Accordingly, the volume of the fundamental domain is 7 and ''K'' is only [[Splitting of prime ideals in Galois extensions|ramified]] at 7.]]
+In [[mathematics]], the '''discriminant of an [[algebraic number field]]''' is a numerical [[invariant (mathematics)|invariant]] that, loosely speaking, measures the size of the ([[ring of integers]] of the) algebraic number field. More specifically, it is related to the volume of the [[fundamental domain]] of the ring of integers, and it regulates which [[prime number|primes]] are [[Ramified prime#In algebraic number theory|ramified]].
+The discriminant is one of the most basic invariants of a number field, and occurs in several important [[Analytic Number Theory|analytic]] formulas such as the [[functional equation (L-function)|functional equation]] of the [[Dedekind zeta function]] of ''K'', and the [[analytic class number formula]] for ''K''. An old theorem of [[Charles Hermite|Hermite]] states that there are only finitely many number fields of bounded discriminant, however determining this quantity is still an [[open problem]], and the subject of current research.<ref>{{harvnb|Cohen|Diaz y Diaz|Olivier|2002}}</ref>
+The discriminant of ''K'' can be referred to as the '''absolute discriminant''' of ''K'' to distinguish it from the '''relative discriminant''' of an [[field extension|extension]] ''K''/''L'' of number fields. The latter is an [[Ideal (ring theory)|ideal]] in the ring of integers of ''L'', and like the absolute discriminant it indicates which primes are ramified in ''K''/''L''. It is a generalization of the absolute discriminant allowing for ''L'' to be bigger than '''Q'''; in fact, when ''L''&nbsp;=&nbsp;'''Q''', the relative discriminant of ''K''/'''Q''' is the [[principal ideal]] of '''Z''' generated by the absolute discriminant of ''K''.
+==Definition==
+Let ''K'' be an algebraic number field, and let ''O''<sub>''K''</sub> be its ring of integers. Let ''b''<sub>1</sub>, ..., ''b''<sub>''n''</sub> be an [[integral basis]] of ''O''<sub>''K''</sub> (i.e. a basis as a [[Module (mathematics)|'''Z'''-module]]), and let {σ<sub>1</sub>, ..., σ<sub>''n''</sub>} be the set of embeddings of ''K'' into the [[complex number]]s (i.e. [[injective]] [[ring homomorphism]]s ''K''&nbsp;→&nbsp;'''C'''). The '''discriminant''' of ''K'' is the [[Square (algebra)|square]] of the [[determinant]] of the ''n'' by ''n'' [[Matrix (mathematics)|matrix]] ''B'' whose (''i'',''j'')-entry is σ<sub>''i''</sub>(''b''<sub>''j''</sub>). Symbolically,
+:<math>\Delta_K=\left(\operatorname{det}\left(\begin{array}{cccc}
+\sigma_1(b_1) & \sigma_1(b_2) &\cdots & \sigma_1(b_n) \\
+\sigma_2(b_1) & \ddots & & \vdots \\
+\vdots & & \ddots & \vdots \\
+\sigma_n(b_1) & \cdots & \cdots & \sigma_n(b_n)
+\end{array}\right)\right)^2.
+</math>
+<br>
+Equivalently, the [[Field trace|trace]] from ''K'' to '''Q''' can be used. Specifically, define the [[Trace form#Trace form and discriminant|trace form]] to be the matrix whose (''i'',''j'')-entry is
+'''Tr'''<sub>''K''/'''Q'''</sub>(''b''<sub>''i''</sub>''b''<sub>''j''</sub>). This matrix equals ''B''<sup>T</sup>''B'', so the discriminant of ''K'' is the determinant of this matrix.
+==Examples==
+*[[Quadratic fields|Quadratic number fields]]: let ''d'' be a [[square-free integer]], then the discriminant of <math>K=\mathbf{Q}(\sqrt{d})</math> is<ref name=MP130/>
+:: <math>\Delta_K=\left\{\begin{array}{ll} d &\text{if }d\equiv 1\pmod 4 \\ 4d &\text{if }d\equiv 2,3\pmod 4. \\\end{array}\right.</math>
+:An integer that occurs as the discriminant of a quadratic number field is called a [[fundamental discriminant]].<ref>Definition 5.1.2 of {{harvnb|Cohen|1993}}</ref>
+*[[Cyclotomic field]]s: let ''n''&nbsp;>&nbsp;2 be an integer, let ζ<sub>''n''</sub> be a [[Root of unity|primitive ''n''th root of unity]], and let ''K''<sub>''n''</sub> = '''Q'''(ζ<sub>''n''</sub>) be the ''n''th cyclotomic field. The discriminant of ''K''<sub>''n''</sub> is given by<ref>Proposition 2.7 of {{harvnb|Washington|1997}}</ref><ref name=MP130>{{citation | first1=Yu. I. | last1=Manin | authorlink1=Yuri I. Manin | first2=A. A. | last2=Panchishkin | title=Introduction to Modern Number Theory | series=Encyclopaedia of Mathematical Sciences | volume=49 | edition=Second | year=2007 | isbn=978-3-540-20364-3 | issn=0938-0396 | zbl=1079.11002 | page=130 }}</ref>
+:: <math>\Delta_{K_n} = (-1)^{\varphi(n)/2} \frac{n^{\varphi(n)}}{\displaystyle\prod_{p|n} p^{\varphi(n)/(p-1)}}</math>
+: where <math>\varphi(n)</math> is [[Euler's totient function]], and the product in the denominator is over primes ''p'' dividing ''n''.
+*Power bases: In the case where the ring of integers has a [[power integral basis]], that is, can be written as ''O''<sub>''K''</sub> = '''Z'''[α], the discriminant of ''K'' is equal to the [[discriminant]] of the [[Minimal polynomial (field theory)|minimal polynomial]] of α. To see this, one can chose the integral basis of ''O''<sub>''K''</sub> to be ''b''<sub>1</sub>&nbsp;=&nbsp;1, ''b''<sub>2</sub>&nbsp;=&nbsp;α, ''b''<sub>3</sub>&nbsp;=&nbsp;''α''<sup>2</sup>, ..., ''b''<sub>''n''</sub>&nbsp;=&nbsp;''α''<sup>''n''&minus;1</sup>. Then, the matrix in the definition is the [[Vandermonde matrix]] associated to α<sub>''i''</sub>&nbsp;=&nbsp;σ<sub>''i''</sub>(α), whose determinant squared is
+:: <math>\prod_{1\leq i<j\leq n}(\alpha_i-\alpha_j)^2</math>
+:which is exactly the definition of the discriminant of the minimal polynomial.
+*Let ''K'' = '''Q'''(α) be the number field obtained by [[Adjunction (field theory)|adjoining]] a [[Root of a function|root]] α of the [[polynomial]] ''x''<sup>3</sup>&nbsp;&minus;&nbsp;''x''<sup>2</sup>&nbsp;&minus;&nbsp;2''x''&nbsp;&minus;&nbsp;8. This is [[Richard Dedekind]]'s original example of a number field whose ring of integers does not possess a power basis. An integral basis is given by {1, α, α(α&nbsp;+&nbsp;1)/2} and the discriminant of ''K'' is &minus;503.<ref>{{harvnb|Dedekind|1878}}, pp. 30–31</ref><ref>{{harvnb|Narkiewicz|2004}}, p. 64</ref>
+*Repeated discriminants: the discriminant of a quadratic field uniquely identifies it, but this is not true, in general, for [[degree of a number field|higher-degree]] number fields. For example, there are two [[isomorphism|non-isomorphic]] [[cubic field]]s of discriminant 3969. They are obtained by adjoining a root of the polynomial {{nobreak|''x''<sup>3</sup> &minus; 21''x'' + 28}} or {{nobreak|''x''<sup>3</sup> &minus; 21''x'' &minus; 35}}, respectively.<ref>{{harvnb|Cohen|1993|loc=Theorem 6.4.6}}</ref>
+<!-- Non-power basis example lacking reference
+*Let ''K'' = '''Q'''(α) be the number field obtained by adjoining a root α of the polynomial ''x''<sup>3</sup>&nbsp;&minus;&nbsp;11''x''<sup>2</sup>&nbsp;+&nbsp;''x''&nbsp;+&nbsp;1. This is an example that does not have a power basis. An integral basis is given by {1, α, 1/2(α<sup>2</sup>&nbsp;+&nbsp;1)}, and the trace form is
+:: <math>\left(\begin{array}{ccc}
+& 11 & 61 \\
+& 119 & 653 \\
+& 653 & 3589 \\
+\end{array}\right).</math>
+: The discriminant of ''K'' is the determinant of this matrix, which is 1304 = 2<sup>3</sup> 163.
+-->
+==Basic results==
+*'''Brill's theorem''':<ref>{{harvnb|Koch|1997|p=11}}</ref> The [[sign (mathematics)|sign]] of the discriminant is (&minus;1)<sup>''r''<sub>2</sub></sup> where ''r''<sub>2</sub> is the number of [[Algebraic number field#Archimedean places|complex places]] of ''K''.<ref>Lemma 2.2 of {{harvnb|Washington|1997}}</ref>
+*A prime ''p'' ramifies in ''K'' if, and only if, ''p'' divides Δ<sub>''K''</sub>&nbsp;.<ref>Corollary III.2.12 of {{harvnb|Neukirch|1999}}</ref>
+*'''Stickelberger's theorem''':<ref>Exercise I.2.7 of {{harvnb|Neukirch|1999}}</ref>
+:: <math>\Delta_K\equiv 0\text{ or }1 \pmod 4.</math>
+*'''[[Minkowski's bound]]''':<ref>Proposition III.2.14 of {{harvnb|Neukirch|1999}}</ref> Let ''n'' denote the [[Degree of a field extension|degree]] of the extension ''K''/'''Q''' and ''r''<sub>2</sub> the number of complex places of ''K'', then
+:: <math>|\Delta_K|^{1/2}\geq \frac{n^n}{n!}\left(\frac{\pi}{4}\right)^{r_2} \geq \frac{n^n}{n!}\left(\frac{\pi}{4}\right)^{n/2}.</math>
+*'''Minkowski's theorem''':<ref>Theorem III.2.17 of {{harvnb|Neukirch|1999}}</ref> If ''K'' is not '''Q''', then |Δ<sub>''K''</sub>| > 1 (this follows directly from the Minkowski bound).
+*'''[[Hermite–Minkowski theorem]]''':<ref>Theorem III.2.16 of {{harvnb|Neukirch|1999}}</ref> Let ''N'' be a positive integer. There are only finitely many (up to isomorphisms) algebraic number fields ''K'' with |Δ<sub>''K''</sub>| &lt; ''N''.  Again, this follows from the Minkowski bound together Hermite's theorem (There are only finitely many algebraic number fields with prescribed discriminant).
+==History==
+[[File:Dedekind.jpeg|thumb|150px|right|Richard Dedekind showed that every number field possesses an integral basis, allowing him to define the discriminant of an arbitrary number field.<ref name=DedekindX>Dedekind's supplement X of the second edition of [[Peter Gustav Lejeune Dirichlet]]'s [[Vorlesungen über Zahlentheorie]] {{harv|Dedekind|1871}}</ref>]]
+The definition of the discriminant of a general algebraic number field, ''K'', was given by Dedekind in 1871.<ref name=DedekindX/> At this point, he already knew the relationship between the discriminant and ramification.<ref>{{harvnb|Bourbaki|1994}}</ref>
+Hermite's theorem predates the general definition of the discriminant with Charles Hermite publishing a proof of it in 1857.{{sfn|Hermite|1857}} In 1877, [[Alexander von Brill]] determined the sign of the discriminant.{{sfn|Brill|1877}} [[Leopold Kronecker]] first stated Minkowski's theorem in 1882,{{sfn|Kronecker|1882}} though the first proof was given by Hermann Minkowski in 1891.{{sfn|Minkowski|1891a}} In the same year, Minkowski published his bound on the discriminant.{{sfn|Minkowski| 1891b}} Near the end of the nineteenth century, [[Ludwig Stickelberger]] obtained his theorem on the residue of the discriminant modulo four.{{sfn|Stickelberger|1897}}<ref>All facts in this paragraph can be found in {{harvnb|Narkiewicz|2004|pp=59, 81}}</ref>
+==<span id"RelativeDiscriminant"></span>Relative discriminant==
+The discriminant defined above is sometimes referred to as the ''absolute'' discriminant of ''K'' to distinguish it from the '''relative discriminant''' Δ<sub>''K''/''L''</sub> of an extension of number fields ''K''/''L'', which is an ideal in ''O''<sub>''L''</sub>. The relative discriminant is defined in a fashion similar to the absolute discriminant, but must take into account that ideals in ''O''<sub>''L''</sub> may not be principal and that there may not be an integral basis of ''K''/''L''. Let {σ<sub>1</sub>, ..., σ<sub>''n''</sub>} be the set of embeddings of ''K'' into '''C''' which are the identity on ''L''. If ''b''<sub>1</sub>, ..., ''b''<sub>''n''</sub> is any basis of ''K'' over ''L'', let ''d''(''b''<sub>1</sub>, ..., ''b''<sub>''n''</sub>) be the square of the determinant of the ''n'' by ''n'' matrix whose (''i'',''j'')-entry is σ<sub>''i''</sub>(''b''<sub>''j''</sub>). Then, the relative discriminant of ''K''/''L'' is the ideal generated by the ''d''(''b''<sub>1</sub>, ..., ''b''<sub>''n''</sub>) as {''b''<sub>1</sub>, ..., ''b''<sub>''n''</sub>} varies over all bases of ''K''/''L'' with the property that ''b<sub>i''</sub>&nbsp;∈&nbsp;''O<sub>K''</sub> for all ''i''. Alternatively, the relative discriminant of ''K''/''L'' is the [[ideal norm|norm]] of the [[different ideal|different]] of ''K''/''L''.<ref name="NeukirchIII2">{{harvnb|Neukirch|1999|loc=§III.2}}</ref> When ''L'' = '''Q''', the relative discriminant Δ<sub>''K''/'''Q'''</sub> is the principal ideal of '''Z''' generated by the absolute discriminant Δ<sub>''K''</sub>&nbsp;. In a [[tower of fields]] ''K''/''L''/''F'' the relative discriminants are related by
+:<math>\Delta_{K/F} = \mathcal{N}_{L/F}\left({\Delta_{K/L}}\right) \Delta_{L/F}^{[K:L]}</math>
+where <math>\mathcal{N}</math> denotes relative [[field norm|norm]].<ref>Corollary III.2.10 of {{harvnb|Neukirch|1999}} or Proposition III.2.15 of {{harvnb|Fröhlich|Taylor|1993}}</ref>
+===Ramification===
+The relative discriminant regulates the [[Tame ramification|ramification]] data of the field extension ''K''/''L''. A prime ideal ''p'' of ''L'' ramifies in ''K'' if, and only if, it divides the relative discriminant Δ<sub>''K''/''L''</sub>. An extension is unramified if, and only if, the discriminant is the unit ideal.<ref name="NeukirchIII2"/> The Minkowski bound above shows that there are no non-trivial unramified extensions of '''Q'''. Fields larger than '''Q''' may have unramified extensions, for example, for any field with [[class number (number theory)|class number]] greater than one, its [[Hilbert class field]] is a non-trivial unramified extension.
+==Root discriminant==
+The '''root discriminant''' of a number field, ''K'', of degree ''n'', often denoted rd<sub>''K''</sub>, is defined as the ''n''-th root of the absolute value of the (absolute) discriminant of ''K''.<ref name=voight2008>{{harvnb|Voight|2008}}</ref> The relation between relative discriminants in a tower of fields shows that the root discriminant does not change in an unramified extension.  The existence of a [[class field tower]] provides bounds on the root discriminant: the existence of an infinite class field tower over '''Q'''(√-m) where ''m'' = 3·5·7·11·19 shows that there are infinitely many fields with root discriminant 2√''m'' ≈ 296.276.<ref name=koch181>{{harvnb|Koch|1997|pp=181–182}}</ref>  If we let ''r'' and 2''s'' be the number of real and complex embeddings, so that ''n''&nbsp;=&nbsp;''r''&nbsp;+&nbsp;2''s'', put ''ρ''&nbsp;=&nbsp;''r''/''n'' and ''σ''&nbsp;=&nbsp;2''s''/''n''.  Set ''α''(''ρ'',&nbsp;''σ'') to be the infimum of rd<sub>''K''</sub> for ''K'' with (''r',&nbsp;2''s') = (''ρn'',&nbsp;''σn'').  We have<ref name=koch181/>
+:<math> \alpha(\rho,\sigma) \ge 60.8^\rho 22.3^\sigma </math>
+and on the assumption of the [[generalized Riemann hypothesis]]
+:<math> \alpha(\rho,\sigma) \ge 215.3^\rho 44.7^\sigma . </math>
+So we have ''α''(0,1)&nbsp;<&nbsp;296.276.  Martinet has shown ''α''(0,1)&nbsp;<&nbsp;93 and ''α''(1,0)&nbsp;<&nbsp;1059.<ref name=koch181/><ref>{{cite journal | zbl=0369.12007 | last=Martinet | first=Jacques | title=Tours de corps de classes et estimations de discriminants | language=French | journal=[[Inventiones Mathematicae]] | volume=44 | pages=65–73 | year=1978 }}</ref>  {{harvnb|Voight|2008}} proves that for totally real fields, the root discriminant is >&nbsp;14, with 1229 exceptions.
+==Relation to other quantities==
+*When embedded into <math>K\otimes_\mathbf{Q}\mathbf{R}</math>, the volume of the fundamental domain of ''O''<sub>''K''</sub> is <math>\sqrt{|\Delta_K|}</math> (sometimes a different [[Measure (mathematics)|measure]] is used and the volume obtained is <math>2^{-r_2}\sqrt{|\Delta_K|}</math>, where ''r''<sub>2</sub> is the number of complex places of ''K'').
+*Due to its appearance in this volume, the discriminant also appears in the functional equation of the Dedekind zeta function of ''K'', and hence in the analytic class number formula, and the [[Brauer–Siegel theorem]].
+*The relative discriminant of ''K''/''L'' is the [[Artin conductor]] of the [[regular representation]] of the [[Galois group]] of ''K''/''L''. This provides a relation to the Artin conductors of the [[character group|characters]] of the Galois group of ''K''/''L'', called the [[conductor-discriminant formula]].<ref>Section 4.4 of {{harvnb|Serre|1967}}</ref>
+==Notes==
+{{reflist|2}}
+==References==
+===Primary sources===
+* {{Citation
+| last=Brill
+| first=Alexander von
+| title=Ueber die Discriminante
+| year=1877
+| journal=Mathematische Annalen
+| volume=12
+| pages=87–89
+| doi=10.1007/BF01442468
+| url=http://gdz.sub.uni-goettingen.de/en/dms/load/toc/?PPN=PPN235181684_0012&DMDID=dmdlog8
+| accessdate=2009-08-22
+| issue=1
+| mr=1509928
+| jfm=09.0059.02
+}}
+* {{Citation
+| last=Dedekind
+| first=Richard
+| author-link=Richard Dedekind
+| title=Vorlesungen über Zahlentheorie von P.G. Lejeune Dirichlet
+| url=http://gdz.sub.uni-goettingen.de/en/dms/load/toc/?PPN=PPN30976923X&DMDID=dmdlog1
+| edition=2
+| year=1871
+| publisher=Vieweg
+| accessdate=2009-08-05
+}}
+* {{Citation
+| last=Dedekind
+| first=Richard
+| author-link=Richard Dedekind
+| title=Über den Zusammenhang zwischen der Theorie der Ideale und der Theorie der höheren Congruenzen
+| url=http://gdz.sub.uni-goettingen.de/en/dms/load/toc/?PPN=PPN250442582_0023&DMDID=dmdlog10
+| year=1878
+| journal=Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen
+| volume=23
+| accessdate=2009-08-20
+| issue=1
+}}
+*{{Citation
+| last=Hermite
+| first=Charles
+| author-link=Charles Hermite
+| title=Extrait d'une lettre de M. C. Hermite à M. Borchardt sur le nombre limité d'irrationalités auxquelles se réduisent les racines des équations à coefficients entiers complexes d'un degré et d'un discriminant donnés
+| journal=[[Crelle's Journal]]
+| volume=53
+| pages=182–192
+| year=1857
+| url=http://gdz.sub.uni-goettingen.de/dms/load/toc/?PPN=PPN243919689_0053&DMDID=dmdlog14
+| accessdate=2009-08-20
+}}
+*{{Citation
+| last=Kronecker
+| first=Leopold
+| author-link=Leopold Kronecker
+| title=Grundzüge einer arithmetischen Theorie der algebraischen Grössen
+| url=http://gdz.sub.uni-goettingen.de/dms/load/toc/?PPN=PPN243919689_0092&DMDID=dmdlog4
+| journal=[[Crelle's journal]]
+| volume=92
+| year=1882
+| pages=1–122
+| accessdate=2009-08-20
+| jfm=14.0038.02
+}}
+*{{Citation
+| last=Minkowski
+| first=Hermann
+| author-link=Hermann Minkowski
+| title=Ueber die positiven quadratischen Formen und über kettenbruchähnliche Algorithmen
+| url=http://gdz.sub.uni-goettingen.de/dms/load/toc/?PPN=PPN243919689_0107&DMDID=dmdlog25
+| journal=Crelle's journal
+| volume=107
+| year=1891a
+| pages=278–297
+| accessdate=2009-08-20
+| jfm=23.0212.01
+}}
+*{{Citation
+| last=Minkowski
+| first=Hermann
+| author-link=Hermann Minkowski
+| title=Théorèmes d'arithmétiques
+| journal=[[Comptes rendus de l'Académie des sciences]]
+| year=1891b
+| volume=112
+| pages=209–212
+| jfm=23.0214.01
+}}
+*{{Citation
+| last=Stickelberger
+| first=Ludwig
+| author-link=Ludwig Stickelberger
+| contribution=Über eine neue Eigenschaft der Diskriminanten algebraischer Zahlkörper
+| title=Proceedings of the First International Congress of Mathematicians, Zürich
+| year=1897
+| pages=182–193
+| jfm=29.0172.03
+}}
+===Secondary sources===
+* {{Bourbaki EHM}}
+* {{Citation
+| last=Cohen
+| first=Henri
+| author-link=Henri Cohen (number theorist)
+| title=A Course in Computational Algebraic Number Theory
+| publisher=[[Springer-Verlag]]
+| location=Berlin, New York
+| series=Graduate Texts in Mathematics
+| isbn=978-3-540-55640-4
+| year=1993
+| volume=138
+| mr=1228206
+}}
+* {{Citation
+| last=Cohen
+| first=Henri
+| author-link=Henri Cohen (number theorist)
+| last2=Diaz y Diaz
+| first2=Francisco
+| last3=Olivier
+| first3=Michel
+| contribution=A Survey of Discriminant Counting
+| contribution-url=http://www.springerlink.com/content/2bl9qep1qb4k32vy/
+| title=Algorithmic Number Theory, Proceedings, 5th International Syposium, ANTS-V, University of Sydney, July 2002
+| editor-last=Fieker
+| editor-first=Claus
+| editor2-last=Kohel
+| editor2-first=David R.
+| publisher=Springer-Verlag
+| location=Berlin
+| series=Lecture Notes in Computer Science
+| issn=0302-9743
+| isbn=978-3-540-43863-2
+| doi=10.1007/3-540-45455-1_7
+| year=2002
+| volume=2369
+| pages=80–94
+| accessdate=2009-08-19
+| mr=2041075
+}}
+* {{Citation
+| last= Fröhlich
+| first = Albrecht
+| authorlink= Albrecht Fröhlich
+| last2=Taylor
+| first2=Martin
+| authorlink2= Martin J. Taylor
+| title=Algebraic number theory
+| publisher=[[Cambridge University Press]]
+| series=Cambridge Studies in Advanced Mathematics
+| isbn=978-0-521-43834-6
+| year=1993
+| volume=27
+| mr= 1215934
+}}
+* {{citation | first=Helmut | last=Koch | title=Algebraic Number Theory | publisher=[[Springer-Verlag]] | year=1997 | isbn=3-540-63003-1 | zbl=0819.11044 | series=Encycl. Math. Sci. | volume=62 | edition=2nd printing of 1st}}
+* {{Citation
+| last=Narkiewicz
+| first=Władysław
+| title=Elementary and analytic theory of algebraic numbers
+| edition=3
+| year=2004
+| publisher=Springer-Verlag
+| location=Berlin
+| series=Springer Monographs in Mathematics
+| isbn=978-3-540-21902-6
+| mr=2078267
+}}
+* {{Neukirch ANT}}
+* {{Citation
+| last=Serre
+| first=Jean-Pierre
+| author-link=Jean-Pierre Serre
+| chapter=Local class field theory
+| title=Algebraic Number Theory, Proceedings of an instructional conference at the University of Sussex, Brighton, 1965
+| editor-last=Cassels
+| editor-first=J. W. S.
+| editor-link=J. W. S. Cassels
+| editor2-last=Fröhlich
+| editor2-first=Albrecht
+| editor2-link=Albrecht Fröhlich
+| publisher=Academic Press
+| location=London
+| isbn=0-12-163251-2
+| year=1967
+| mr=0220701
+}}
+* {{Citation
+| first = John
+| last = Voight
+| contribution = Enumeration of totally real number fields of bounded root discriminant
+| title = Algorithmic number theory. Proceedings, 8th International Symposium, ANTS-VIII, Banff, Canada, May 2008
+| editor1-last=van der Poorten | editor1-first=Alfred J. | editor1-link=Alfred van der Poorten
+| editor2-last=Stein
+| editor2-first=Andreas
+| pages = 268–281
+| publisher = Springer-Verlag
+| location=Berlin
+| series=Lecture Notes in Computer Science
+| volume=5011
+| year = 2008
+| arxiv = 0802.0194
+| doi = 10.1007/978-3-540-79456-1_18
+| isbn=978-3-540-79455-4| mr = 2467853 | zbl=1205.11125
+}}
+* {{Citation
+| last=Washington
+| first=Lawrence
+| title=Introduction to Cyclotomic Fields
+| edition=2 nd
+| publisher=Springer-Verlag
+| location=Berlin, New York
+| series=Graduate Texts in Mathematics
+| isbn=978-0-387-94762-4
+| year=1997
+| volume=83
+| mr=1421575 | zbl=0966.11047
+}}
+==Further reading==
+* {{Citation | last=Milne | first=James S. | author-link=James S. Milne | title=Algebraic Number Theory | year=1998 | url=http://www.jmilne.org/math/CourseNotes/ant.html | accessdate=2008-08-20}}
+{{DEFAULTSORT:Discriminant Of An Algebraic Number Field}}
+[[Category:Algebraic number theory]]

Direct image functor: Difference between revisions

Latest revision as of 16:29, 11 November 2013

Definition

Examples

Basic results

History

Relative discriminant

Ramification

Root discriminant

Relation to other quantities

Notes

References

Primary sources

Secondary sources

Further reading

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