Equated Monthly Installment

In algebraic number theory Eisenstein's reciprocity law is a reciprocity law that extends the law of quadratic reciprocity and the cubic reciprocity law to residues of higher powers. It is one of the earliest and simplest of the higher reciprocity laws, and is a consequence of several later and stronger reciprocity laws such as the Artin reciprocity law. It was introduced by Template:Harvs, though Jacobi had previously announced (without proof) a similar result for the special cases of 5th, 8th and 12th powers in 1839.^[1]

Background and notation

Let $m > 1$ be an integer, and let $𝒪_{m}$ be the ring of integers of the m-th cyclotomic field $ℚ (ζ_{m}),$ where $ζ_{m} = e^{2 π i \frac{1}{m}}$ is a primitive m-th root of unity.

The numbers $ζ_{m}, ζ_{m}^{2}, \dots ζ_{m}^{m} = 1$ are units in $𝒪_{m} .$ (There are other units as well.)

Primary numbers

A number $α \in 𝒪_{m}$ is called primary^[2]^[3] if it is not a unit, is relatively prime to $m$ , and is congruent to a rational (i.e. in $ℤ$ ) integer $(\mod (1 - ζ_{m})^{2}) .$

The following lemma^[4]^[5] shows that primary numbers in $𝒪_{m}$ are analogous to positive integers in $ℤ .$

Suppose that $α, β \in 𝒪_{m}$ and that both $α$ and $β$ are relatively prime to $m .$ Then

There is an integer $c$ making $ζ_{m}^{c} α$ primary. This integer is unique $(\mod m) .$
if $α$ and $β$ are primary then $α \pm β$ is primary.
if $α$ and $β$ are primary then $α β$ is primary.>
$α^{m}$ is primary.

The significance of $1 - ζ_{m}$ which appears in the definition is most easily seen when $m = l$ is a prime. In that case $l = (1 - ζ_{l}) (1 - ζ_{l}^{2}) \dots (1 - ζ_{l}^{l - 1}) .$ Furthermore, the prime ideal $(l)$ of $ℤ$ is totally ramified in $ℚ (ζ_{l})$

$(l) = (1 - ζ_{l})^{l - 1},$

and the ideal $(1 - ζ_{l})$ is prime of degree 1.^[6]^[7]

m-th power residue symbol

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For $α, β \in 𝒪_{m},$ the m-th power residue symbol for $𝒪_{m}$ is either zero or an m-th root of unity:

{(\frac{α}{β})}_{m} = {\begin{cases} ζ where ζ^{m} = 1 & if α and β are relatively prime \\ 0 & otherwise . \end{cases}

It is the m-th power version of the classical (quadratic, m = 2) Jacobi symbol (assuming $α$ and $β$ are relatively prime):

If $η \in 𝒪_{m}$ and $α \equiv η^{m} (\mod β)$ then ${(\frac{α}{β})}_{m} = 1 .$

If ${(\frac{α}{β})}_{m} \neq 1$ then $α$ is not an m-th power $(\mod β) .$

If ${(\frac{α}{β})}_{m} = 1$ then $α$ may or may not be an m-th power $(\mod β) .$

Statement of the theorem

Let $m \in ℤ$ be an odd prime and $a \in ℤ$ an integer relatively prime to $m .$ Then

First supplement

{(\frac{ζ_{m}}{a})}_{m} = ζ_{m}^{\frac{a^{m - 1} - 1}{m}} .

^[8]

Second supplement

{(\frac{1 - ζ_{m}}{a})}_{m} = {(\frac{ζ_{m}}{a})}_{m}^{\frac{m - 1}{2}} .

^[8]

Eisenstein reciprocity

Let $α \in 𝒪_{m}$ be primary (and therefore relatively prime to $m$ ), and assume that $α$ is also relatively prime to $a$ Then

{(\frac{α}{a})}_{m} = {(\frac{a}{α})}_{m} .

^[9]^[8]

Proof

The theorem is a consequence of the Stickelberger relation.^[10]^[11]

Template:Harvtxt gives a historical discussion of some early reciprocity laws, including a proof of Eisenstein's law using Gauss and Jacobi sums that is based on Eisenstein's original proof..

Generalization

In 1922 Takagi proved that if $K \supset ℚ (ζ_{l})$ is an arbitrary algebraic number field containing the $l$ -th roots of unity for a prime $l$ , then Eisenstein's law for $l$ -th powers holds in $K .$ ^[12]

Applications

First case of Fermat's last theorem

Assume that $p$ is an odd prime, that $x^{p} + y^{p} + z^{p} = 0$ for pairwise relatively prime integers (i.e. in $ℤ$ ) $x, y, z$ and that $p ∤ x y z .$

This is the first case of Fermat's last theorem. (The second case is when $p ∣ x y z .$ ) Eisenstein reciprocity can be used to prove the following theorems

(Wieferich 1909)^[13]^[14] Under the above assumptions, $2^{p - 1} \equiv 1 (\mod p^{2}) .$

The only primes below 6.7×10¹⁵ that satisfy this are 1093 and 3511. See Wieferich primes for details and current records.

(Mirimanoff 1911)^[15] Under the above assumptions $3^{p - 1} \equiv 1 (\mod p^{2}) .$

Analogous results are true for all primes ≤ 113, but the proof does not use Eisenstein's law. See Wieferich prime#Connection with Fermat's last theorem.

(Furtwängler 1912)^[16]^[17] Under the above assumptions, for every prime $r ∣ x, r^{p - 1} \equiv 1 (\mod p^{2}) .$

(Furtwängler 1912)^[18] Under the above assumptions, for every prime $r ∣ (x - y), r^{p - 1} \equiv 1 (\mod p^{2}) .$

(Vandiver)^[19] Under the above assumptions, if in addition $p > 3,$ then $x^{p} \equiv x, y^{p} \equiv y$ and $z^{p} \equiv z (\mod p^{3}) .$

Powers mod most primes

Eisenstein's law can be used to prove the following theorem (Trost, Ankeny, Rogers).^[20] Suppose $a \in ℤ$ and that $l ∤ a$ where $l$ is an odd prime. If $x^{l} \equiv a (\mod p)$ is solvable for all but finitely many primes $p$ then $a = b^{l} .$

Notes

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References

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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

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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

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

↑ Lemmermeyer, p. 392.
↑ Ireland & Rosen, ch. 14.2
↑ Lemmermeyer, ch. 11.2, uses the term semi-primary.
↑ Ireland & Rosen, lemma in ch. 14.2 (first assertion only)
↑ Lemmereyer, lemma 11.6
↑ Ireland & Rosen, prop 13.2.7
↑ Lemmermeyer, prop. 3.1
↑ ^8.0 ^8.1 ^8.2 Lemmermeyer, thm. 11.9
↑ Ireland & Rosen, ch. 14 thm. 1
↑ Ireland & Rosen, ch. 14.5
↑ Lemmermeyer, ch. 11.2
↑ Lemmermeyer, ch. 11 notes
↑ Lemmermeyer, ex. 11.33
↑ Ireland & Rosen, th. 14.5
↑ Lemmermeyer, ex. 11.37
↑ Lemmermeyer, ex. 11.32
↑ Ireland & Rosen, th. 14.6
↑ Lemmermeyer, ex. 11.36
↑ Ireland & Rosen, notes to ch. 14
↑ Ireland & Rosen, ch. 14.6, thm. 4. This is part of a more general theorem: Assume $x^{n} \equiv a (\mod p)$ for all but finitely many primes $p .$ Then i) if $8 ∤ n$ then $a = b^{n}$ but ii) if $8 | n$ then $a = b^{n}$ or $a = 2^{\frac{n}{2}} b^{n} .$

[1] Lemmermeyer, p. 392.

[2] Ireland & Rosen, ch. 14.2

[3] Lemmermeyer, ch. 11.2, uses the term semi-primary.

[4] Ireland & Rosen, lemma in ch. 14.2 (first assertion only)

[5] Lemmereyer, lemma 11.6

[6] Ireland & Rosen, prop 13.2.7

[7] Lemmermeyer, prop. 3.1

[Lemmermeyer,_thm._11.9-8] 8.0 ^8.1 ^8.2 Lemmermeyer, thm. 11.9

[9] Ireland & Rosen, ch. 14 thm. 1

[10] Ireland & Rosen, ch. 14.5

[11] Lemmermeyer, ch. 11.2

[12] Lemmermeyer, ch. 11 notes

[13] Lemmermeyer, ex. 11.33

[14] Ireland & Rosen, th. 14.5

[15] Lemmermeyer, ex. 11.37

[16] Lemmermeyer, ex. 11.32

[17] Ireland & Rosen, th. 14.6

[18] Lemmermeyer, ex. 11.36

[19] Ireland & Rosen, notes to ch. 14

[20] Ireland & Rosen, ch. 14.6, thm. 4. This is part of a more general theorem: Assume $x^{n} \equiv a (\mod p)$ for all but finitely many primes $p .$ Then i) if $8 ∤ n$ then $a = b^{n}$ but ii) if $8 | n$ then $a = b^{n}$ or $a = 2^{\frac{n}{2}} b^{n} .$

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Equated Monthly Installment

Contents

Background and notation

Primary numbers

m-th power residue symbol

Statement of the theorem

First supplement

Second supplement

Eisenstein reciprocity

Proof

Generalization

Applications

First case of Fermat's last theorem

Powers mod most primes

See also

Notes

References

Navigation menu

Equated Monthly Installment

Background and notation

Primary numbers

m-th power residue symbol

Statement of the theorem

First supplement

Second supplement

Eisenstein reciprocity

Proof

Generalization

Applications

First case of Fermat's last theorem

Powers mod most primes

See also

Notes

References

Navigation menu

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