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In [[mathematics]], the '''Dedekind [[zeta function]]''' of an [[algebraic number field]] ''K'', generally denoted ζ<sub>''K''</sub>(''s''), is a generalization of the [[Riemann zeta function]]&mdash;which is obtained by specializing to the case where ''K'' is the [[rational number]]s '''Q'''. In particular, it can be defined as a [[Dirichlet series]], it has an [[Euler product]] expansion, it satisfies a [[functional equation (L-function)|functional equation]], it has an [[analytic continuation]] to a [[meromorphic function]] on the [[complex plane]] '''C''' with only a [[simple pole]] at ''s''&nbsp;=&nbsp;1, and its values encode arithmetic data of ''K''. The [[extended Riemann hypothesis]] states that if ''ζ''<sub>''K''</sub>(''s'')&nbsp;=&nbsp;0 and 0&nbsp;<&nbsp;Re(''s'')&nbsp;<&nbsp;1, then Re(''s'')&nbsp;=&nbsp;1/2.
{{mergefrom|Conjugate (algebra)|date=January 2012}}
In [[mathematics]], the '''difference of two squares''', or the difference of perfect squares, is a [[Square (algebra)|squared]] (multiplied by itself) number subtracted from another squared number. It refers to the [[identity (mathematics)|identity]]


:<math>a^2-b^2 = (a+b)(a-b)\,\!</math>
The Dedekind zeta function is named for [[Richard Dedekind]] who introduced them in his supplement to [[Peter Gustav Lejeune Dirichlet]]'s [[Vorlesungen über Zahlentheorie]].<ref>{{harvnb|Narkiewicz|2004|loc=§7.4.1}}</ref>


in [[elementary algebra]].
==Definition and basic properties==
Let ''K'' be an [[algebraic number field]]. Its Dedekind zeta function is first defined for complex numbers ''s'' with [[real part]] Re(''s'')&nbsp;>&nbsp;1 by the Dirichlet series


==Proof==
:<math>\zeta_K (s) = \sum_{I \subseteq \mathcal{O}_K} \frac{1}{(N_{K/\mathbf{Q}} (I))^{s}}</math>
The [[mathematical proof|proof]] is straightforward. Starting from the [[Sides of an equation|right-hand side]], apply the [[distributive law]] to get
:<math>(a+b)(a-b) = a^2+ba-ab-b^2\,\!</math>,
and set
:<math>ba - ab = 0\,\!</math>
as an application of the [[commutative law]]. The resulting identity is one of the most commonly used in mathematics.


The proof just given indicates the scope of the identity in [[abstract algebra]]: it will hold in any [[commutative ring]] ''R''.
where ''I'' ranges through the non-zero [[ideal (ring theory)|ideals]] of the [[ring of integers]] ''O''<sub>''K''</sub> of ''K'' and ''N''<sub>''K''/'''Q'''</sub>(''I'') denotes the [[absolute norm]] of ''I'' (which is equal to both the [[Index of a subgroup|index]] [''O''<sub>''K''</sub>&nbsp;:&nbsp;''I''] of ''I'' in ''O''<sub>''K''</sub> or equivalently the [[cardinality]] of [[quotient ring]] ''O''<sub>''K''</sub>&nbsp;/&nbsp;''I''). This sum converges absolutely for all complex numbers ''s'' with [[real part]] Re(''s'')&nbsp;>&nbsp;1. In the case ''K''&nbsp;=&nbsp;'''Q''', this definition reduces to that of the Riemann zeta function.


Conversely, if this identity holds in a [[ring (mathematics)|ring]] ''R'' for all pairs of elements ''a'' and ''b'' of the ring, then ''R'' is commutative.  To see this, we apply the distributive law to the right-hand side of the original equation and get
===Euler product===
:<math>a^2 + ba - ab - b^2\,\!</math>
The Dedekind zeta function of ''K'' has an Euler product which is a product over all the [[prime ideal]]s ''P'' of ''O''<sub>''K''</sub>


and for this to be equal to <math>a^2 - b^2</math>, we must have
:<math>\zeta_K (s) = \prod_{P \subseteq \mathcal{O}_K} \frac{1}{1 - (N_{K/\mathbf{Q}}(P))^{-s}},\text{ for Re}(s)>1.</math>


:<math>ba - ab = 0\,\!</math>
This is the expression in analytic terms of the [[Dedekind domain|uniqueness of prime factorization of the ideals]] ''I'' in ''O''<sub>''K''</sub>. The fact that, for Re(''s'')&nbsp;>&nbsp;1, ζ<sub>''K''</sub>(''s'') is given by a product of non-zero numbers implies that it is non-zero in this region.


for all pairs ''a'', ''b'' of elements of ''R'', so the ring ''R'' is commutative.
===Analytic continuation and functional equation===
[[Erich Hecke]] first proved that ''ζ''<sub>''K''</sub>(''s'') has an analytic continuation to the complex plane as a meromorphic function, having a simple pole only at ''s''&nbsp;=&nbsp;1. The [[Residue (complex analysis)|residue]] at that pole is given by the [[analytic class number formula]] and is made up of important arithmetic data involving invariants of the [[unit group]] and [[class group]] of ''K''.


==In geometry==
The Dedekind zeta function satisfies a functional equation relating its values at ''s'' and 1&nbsp;&minus;&nbsp;''s''. Specifically, let Δ<sub>''K''</sub> denote [[Discriminant of an algebraic number field|discriminant]] of ''K'', let ''r''<sub>1</sub> (resp. ''r''<sub>2</sub>) denote the number of [[real place]]s (resp. [[complex place]]s) of ''K'', and let
[[Image:Difference of two squares.png|right]]


The difference of two squares can also be illustrated geometrically as the difference of two square areas in a [[Plane (mathematics)|plane]]. In the diagram, the shaded part represents the difference between the areas of the two squares, i.e. <math>a^2 - b^2</math>.  The area of the shaded part can be found by adding the areas of the two rectangles; <math>a(a-b) + b(a-b)</math>, which can be factorized to <math>(a+b)(a-b)</math>.  Therefore <math>a^2 - b^2 = (a+b)(a-b)</math>
:<math>\Gamma_\mathbf{R}(s)=\pi^{-s/2}\Gamma(s/2)</math>


Another geometric proof proceeds as follows: We start with the figure shown in the first diagram below, a large square with a smaller square removed from it. The side of the entire square is a, and the side of the small removed square is b. The area of the shaded region is <math>a^2-b^2</math>. A cut is made, splitting the region into two rectangular pieces, as shown in the second diagram. The larger piece, at the top, has width a and height a-b. The smaller piece, at the bottom, has width a-b and height b. Now the smaller piece can be detached, rotated, and placed to the right of the larger piece. In this new arrangement, shown in the last diagram below, the two pieces together form a rectangle, whose width is <math>a+b</math> and whose height is <math>a-b</math>. This rectangle's area is <math>(a+b)(a-b)</math>. Since this rectangle came from rearranging the original figure, it must have the same area as the original figure. Therefore, <math>a^2-b^2 = (a+b)(a-b)</math>.Any odd number can be expressed as difference of two squares.
and
[[Image:Difference of two squares geometric proof.png]]


==Uses==
:<math>\Gamma_\mathbf{C}(s)=2(2\pi)^{-s}\Gamma(s)</math>
===Complex number case: sum of two squares===
The difference of two squares is used to find the linear factors of the ''sum'' of two squares, using [[complex number]] coefficients.


For example, the root of <math>z^2 + 5\,\!</math> can be found using difference of two squares:
where Γ(''s'') is the [[Gamma function]]. Then, the function


:<math>z^2 + 5\,\!</math>
:<math>\Lambda_K(s)=\left|\Delta_K\right|^{s/2}\Gamma_\mathbf{R}(s)^{r_1}\Gamma_\mathbf{C}(s)^{r_2}\zeta_K(s)</math>
:<math> = z^2 - (\sqrt{-5})^2</math>
:<math> = z^2 - (i\sqrt5)^2</math>
:<math> = (z + i\sqrt5)(z - i\sqrt5)</math>


Therefore the linear factors are <math>(z + i\sqrt5)</math> and <math>(z - i\sqrt5)</math>.
satisfies the functional equation


Since the two factors found by this method are [[Complex conjugate]]s, we can use this in reverse as a method of multiplying a complex number to get a real number. This is used to get real denominators in complex fractions.<ref>[http://www.themathpage.com/alg/complex-numbers.htm#conjugates Complex or imaginary numbers] TheMathPage.com, retrieved 22 December 2011</ref>
:<math>\Lambda_K(s)=\Lambda_K(1-s).\;</math>


===Rationalising denominators===
==Special values==
The difference of two squares can also be used in the [[Rationalisation (mathematics)|rationalising]] of [[irrational number|irrational]] [[denominator]]s.<ref>[http://www.themathpage.com/alg/multiply-radicals.htm Multiplying Radicals] TheMathPage.com, retrieved 22 December 2011</ref> This is a method for removing [[Nth root|surds]] from expressions (or at least moving them), applying to division by some combinations involving [[square root]]s.
Analogously to the Riemann zeta function, the values of the Dedekind zeta function at integers encode (at least conjecturally) important arithmetic data of the field ''K''. For example, the analytic class number formula relates the residue at ''s''&nbsp;=&nbsp;1 to the [[class number (number theory)|class number]] ''h''(''K'') of ''K'', the [[regulator of an algebraic number field|regulator]] ''R''(''K'') of ''K'', the number ''w''(''K'') of roots of unity in ''K'', the absolute discriminant of ''K'', and the number of real and complex places of ''K''. Another example is at ''s''&nbsp;=&nbsp;0 where it has a zero whose order ''r'' is equal to the [[rank of an abelian group|rank]] of the unit group of ''O''<sub>''K''</sub> and the leading term is given by


For example:
:<math>\lim_{s\rightarrow0}s^{-r}\zeta_K(s)=-\frac{h(K)R(K)}{w(K)}.</math>
The denominator of <math>\dfrac{5}{\sqrt{3} + 4}\,\!</math> can be rationalised as follows:


:<math>\dfrac{5}{\sqrt{3} + 4}\,\!</math>
Combining the functional equation and the fact that Γ(''s'') is infinite at all integers less than or equal to zero yields that ''ζ''<sub>''K''</sub>(''s'') vanishes at all negative even integers. It even vanishes at all negative odd integers unless ''K'' is [[totally real number field|totally real]] (i.e. ''r''<sub>2</sub>&nbsp;=&nbsp;0; e.g. '''Q''' or a [[real quadratic field]]). In the totally real case, [[Carl Ludwig Siegel]] showed that ''ζ''<sub>''K''</sub>(''s'') is a non-zero rational number at negative odd integers. [[Stephen Lichtenbaum]] conjectured specific values for these rational numbers in terms of the [[algebraic K-theory]] of ''K''.


:<math> = \dfrac{5}{\sqrt{3} + 4} \times \dfrac{\sqrt{3} - 4}{\sqrt{3} - 4}\,\!</math>
==Relations to other ''L''-functions==
For the case in which ''K'' is an [[abelian extension]] of '''Q''', its Dedekind zeta function can be written as a product of [[Dirichlet L-function]]s. For example, when ''K'' is a [[quadratic field]] this shows that the ratio


:<math> = \dfrac{5(\sqrt{3} - 4)}{(\sqrt{3} + 4)(\sqrt{3} - 4)}\,\!</math>
:<math>\frac{\zeta_K(s)}{\zeta_{\mathbf{Q}}(s)}</math>


:<math> = \dfrac{5(\sqrt{3} - 4)}{\sqrt{3}^2 - 4^2}\,\!</math>
is the ''L''-function ''L''(''s'',&nbsp;χ), where χ is a [[Jacobi symbol]] used as [[Dirichlet character]]. That the zeta function of a quadratic field is a product of the Riemann zeta function and a certain Dirichlet ''L''-function is an analytic formulation of the [[quadratic reciprocity]] law of Gauss.


:<math> = \dfrac{5(\sqrt{3} - 4)}{3 - 16}\,\!</math>
In general, if ''K'' is a [[Galois extension]] of '''Q''' with [[Galois group]] ''G'', its Dedekind zeta function is the [[Artin L-function|Artin ''L''-function]] of the [[regular representation]] of ''G'' and hence has a factorization in terms of Artin ''L''-functions of [[irreducible representation|irreducible]] [[Artin representation]]s of ''G''.


:<math> = -\dfrac{5(\sqrt{3} - 4)}{13}.\,\!</math>
The relation with Artin L-functions shows that if ''L''/''K'' is a Galois extension then <math>\frac{\zeta_L(s)}{\zeta_K(s)}</math> is holomorphic (<math>\zeta_K(s)</math> "divides" <math>\zeta_L(s)</math>): for general extensions the result would follow from the [[Artin conjecture (L-functions)|Artin conjecture for L-functions]].<ref name=Mar19>Martinet (1977) p.19</ref>


Here, the irrational denominator <math>\sqrt{3} + 4\,\!</math> has been rationalised to <math>13\,\!</math>.
Additionally, ''ζ''<sub>''K''</sub>(''s'') is the [[Hasse–Weil zeta function]] of [[Spectrum of a ring|Spec]] ''O''<sub>''K''</sub><ref>{{harvnb|Deninger|1994|loc=§1}}</ref> and the [[motivic L-function|motivic ''L''-function]] of the [[motive (algebraic geometry)|motive]] coming from the [[cohomology]] of Spec ''K''.<ref>{{harvnb|Flach|2004|loc=§1.1}}</ref>
Any odd number can be expressed as difference of two squares.


===Mental Arithmetic===
==Arithmetically equivalent fields==
The difference of two squares can also be used as a arithmetical short cut.  If you are multiplying two numbers whose average is a number which is easily squared the difference of two squares can be used to give you the product of the original two numbers.


For example: <math> 27 \times 33 = (30 - 3)(30 + 3) \!</math>
Two fields are called arithmetically equivalent if they have the same Dedekind zeta function. {{harvs|txt | last1=Bosma | first1=Wieb | last2=de Smit | first2=Bart | year=2002 | volume=2369 }} used [[Gassmann triple]]s to give some examples of pairs of non-isomorphic fields that are arithmetically equivalent. In particular some of these pairs have different class numbers, so the Dedekind zeta function of a number field does not determine its class number.


Which means by using the difference of two squares <math> 27 \times 33 \!</math> can be restated as
==Notes==
{{reflist}}
<math> a^2 - b^2 </math> which is <math>30^2 - 3^2 = 891. \!</math>


==See also==
==References==
*[[Conjugate (algebra)]]


==References==
*{{Citation | last1=Bosma | first1=Wieb | last2=de Smit | first2=Bart | editor1-last=Kohel | editor1-first=David R. | editor2-last=Fieker | editor2-first=Claus | title=Algorithmic number theory (Sydney, 2002) | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Lecture Notes in Comput. Sci. | isbn=978-3-540-43863-2  | doi=10.1007/3-540-45455-1_6 | mr=2041074  | year=2002 | volume=2369 | chapter=On arithmetically equivalent number fields of small degree | pages=67–79}}
{{reflist}}
*Section 10.5.1 of {{Citation
| last=Cohen
| first=Henri
| author-link=Henri Cohen (number theorist)
| title=Number theory, Volume II: Analytic and modern tools
| publisher=Springer
| location=New York
| series=[[Graduate Texts in Mathematics]]
| volume=240
| year=2007
| isbn=978-0-387-49893-5
| mr=2312338
| doi=10.1007/978-0-387-49894-2
}}
*{{Citation
| last=Deninger
| first=Christopher
| contribution=''L''-functions of mixed motives
| title=Motives, Part 1
| series=Proceedings of Symposia in Pure Mathematics
| publisher=[[American Mathematical Society]]
| volume=55.1
| year=1994
| pages=517–525
| editor-last=Jannsen
| editor-first=Uwe
| editor2-last=Kleiman
| editor2-first=Steven
| editor3-last=Serre
| editor3-first=Jean-Pierre
| editor3-link=Jean-Pierre Serre
| isbn=978-0-8218-1635-6
| url=http://wwwmath.uni-muenster.de/u/deninger/about/publikat/cd22.ps
}}
*{{Citation
| last=Flach
| first=Mathias
| contribution=The equivariant Tamagawa number conjecture: a survey
| url=http://www.math.caltech.edu/papers/baltimore-final.pdf
| title=Stark's conjectures: recent work and new directions
| publisher=[[American Mathematical Society]]
| series=Contemporary Mathematics
| volume=358
| pages=79–125
| isbn=978-0-8218-3480-0
| editor-last=Burns
| editor-first=David
| editor2-last=Popescu
| editor2-first=Christian
| editor3-last=Sands
| editor3-first=Jonathan
| editor4-last=Solomon
| editor4-first=David
}}
*{{citation | last=Martinet | first=J. | chapter=Character theory and Artin L-functions | pages=1-87 | title=Algebraic Number Fields, Proc. Symp. London Math. Soc., Univ. Durham 1975 | editor1-last=Fröhlich | editor1-first=A. | editor1-link=Albrecht Fröhlich | publisher=Academic Press | year=1977 | isbn=0-12-268960-7 | zbl=0359.12015 }}
*{{Citation
| last=Narkiewicz
| first=Władysław
| title=Elementary and analytic theory of algebraic numbers
| edition=3 | at=Chapter 7
| year=2004
| publisher=Springer-Verlag
| location=Berlin
| series=Springer Monographs in Mathematics
| isbn=978-3-540-21902-6
| mr=2078267
}}


{{DEFAULTSORT:Difference Of Two Squares}}
{{L-functions-footer}}
[[Category:Elementary algebra]]
[[Category:Mathematical identities]]
[[Category:Articles containing proofs]]


[[ar:فرق مربعي عددين]]
[[Category:Zeta and L-functions]]
[[sv:Konjugatregeln]]
[[Category:Algebraic number theory]]
[[zh:平方差]]

Revision as of 06:33, 13 August 2014

In mathematics, the Dedekind zeta function of an algebraic number field K, generally denoted ζK(s), is a generalization of the Riemann zeta function—which is obtained by specializing to the case where K is the rational numbers Q. In particular, it can be defined as a Dirichlet series, it has an Euler product expansion, it satisfies a functional equation, it has an analytic continuation to a meromorphic function on the complex plane C with only a simple pole at s = 1, and its values encode arithmetic data of K. The extended Riemann hypothesis states that if ζK(s) = 0 and 0 < Re(s) < 1, then Re(s) = 1/2.

The Dedekind zeta function is named for Richard Dedekind who introduced them in his supplement to Peter Gustav Lejeune Dirichlet's Vorlesungen über Zahlentheorie.[1]

Definition and basic properties

Let K be an algebraic number field. Its Dedekind zeta function is first defined for complex numbers s with real part Re(s) > 1 by the Dirichlet series

ζK(s)=∑I⊆𝒪K1(NK/Q(I))s

where I ranges through the non-zero ideals of the ring of integers OK of K and NK/Q(I) denotes the absolute norm of I (which is equal to both the index [OK : I] of I in OK or equivalently the cardinality of quotient ring OK / I). This sum converges absolutely for all complex numbers s with real part Re(s) > 1. In the case K = Q, this definition reduces to that of the Riemann zeta function.

Euler product

The Dedekind zeta function of K has an Euler product which is a product over all the prime ideals P of OK

ζK(s)=∏P⊆𝒪K11−(NK/Q(P))−s, for Re(s)>1.

This is the expression in analytic terms of the uniqueness of prime factorization of the ideals I in OK. The fact that, for Re(s) > 1, ζK(s) is given by a product of non-zero numbers implies that it is non-zero in this region.

Analytic continuation and functional equation

Erich Hecke first proved that ζK(s) has an analytic continuation to the complex plane as a meromorphic function, having a simple pole only at s = 1. The residue at that pole is given by the analytic class number formula and is made up of important arithmetic data involving invariants of the unit group and class group of K.

The Dedekind zeta function satisfies a functional equation relating its values at s and 1 − s. Specifically, let ΔK denote discriminant of K, let r1 (resp. r2) denote the number of real places (resp. complex places) of K, and let

ΓR(s)=π−s/2Γ(s/2)

and

ΓC(s)=2(2π)−sΓ(s)

where Γ(s) is the Gamma function. Then, the function

ΛK(s)=|ΔK|s/2ΓR(s)r1ΓC(s)r2ζK(s)

satisfies the functional equation

ΛK(s)=ΛK(1−s).

Special values

Analogously to the Riemann zeta function, the values of the Dedekind zeta function at integers encode (at least conjecturally) important arithmetic data of the field K. For example, the analytic class number formula relates the residue at s = 1 to the class number h(K) of K, the regulator R(K) of K, the number w(K) of roots of unity in K, the absolute discriminant of K, and the number of real and complex places of K. Another example is at s = 0 where it has a zero whose order r is equal to the rank of the unit group of OK and the leading term is given by

lims→0s−rζK(s)=−h(K)R(K)w(K).

Combining the functional equation and the fact that Γ(s) is infinite at all integers less than or equal to zero yields that ζK(s) vanishes at all negative even integers. It even vanishes at all negative odd integers unless K is totally real (i.e. r2 = 0; e.g. Q or a real quadratic field). In the totally real case, Carl Ludwig Siegel showed that ζK(s) is a non-zero rational number at negative odd integers. Stephen Lichtenbaum conjectured specific values for these rational numbers in terms of the algebraic K-theory of K.

Relations to other L-functions

For the case in which K is an abelian extension of Q, its Dedekind zeta function can be written as a product of Dirichlet L-functions. For example, when K is a quadratic field this shows that the ratio

ζK(s)ζQ(s)

is the L-function L(s, χ), where χ is a Jacobi symbol used as Dirichlet character. That the zeta function of a quadratic field is a product of the Riemann zeta function and a certain Dirichlet L-function is an analytic formulation of the quadratic reciprocity law of Gauss.

In general, if K is a Galois extension of Q with Galois group G, its Dedekind zeta function is the Artin L-function of the regular representation of G and hence has a factorization in terms of Artin L-functions of irreducible Artin representations of G.

The relation with Artin L-functions shows that if L/K is a Galois extension then ζL(s)ζK(s) is holomorphic (ζK(s) "divides" ζL(s)): for general extensions the result would follow from the Artin conjecture for L-functions.[2]

Additionally, ζK(s) is the Hasse–Weil zeta function of Spec OK[3] and the motivic L-function of the motive coming from the cohomology of Spec K.[4]

Arithmetically equivalent fields

Two fields are called arithmetically equivalent if they have the same Dedekind zeta function. Template:Harvs used Gassmann triples to give some examples of pairs of non-isomorphic fields that are arithmetically equivalent. In particular some of these pairs have different class numbers, so the Dedekind zeta function of a number field does not determine its class number.

Notes

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References

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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.

    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

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  1. ↑ Template:Harvnb
  2. ↑ Martinet (1977) p.19
  3. ↑ Template:Harvnb
  4. ↑ Template:Harvnb
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