Lindley's paradox: Difference between revisions

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In mathematics, the Hilbert symbol or norm-residue symbol is a function (–, –) from K^× × K^× to the group of nth roots of unity in a local field K such as the fields of reals or p-adic numbers . It is related to reciprocity laws, and can be defined in terms of the Artin symbol of local class field theory. The Hilbert symbol was introduced by Template:Harvs in his Zahlbericht, with the slight difference that he defined it for elements of global fields rather than for the larger local fields.

The Hilbert symbol has been generalized to higher local fields.

Quadratic Hilbert symbol

Over a local field K whose multiplicative group of non-zero elements is K^×, the quadratic Hilbert symbol is the function (–, –) from K^× × K^× to {−1,1} defined by

(a, b) = {\begin{cases} 1, & if z^{2} = a x^{2} + b y^{2} has a non-zero solution (x, y, z) \in K^{3}; \\ - 1, & if  not. \end{cases}

Properties

The following three properties follow directly from the definition, by choosing suitable solutions of the diophantine equation above:

If a is a square, then (a, b) = 1 for all b.
For all a,b in K^×, (a, b) = (b, a).
For any a in K^× such that a−1 is also in K^×, we have (a, 1−a) = 1.

The (bi)multiplicativity, i.e.,

(a, b₁b₂) = (a, b₁)·(a, b₂)

for any a, b₁ and b₂ in K^× is, however, more difficult to prove, and requires the development of local class field theory.

The third property shows that the Hilbert symbol is an example of a Steinberg symbol and thus factors over the second Milnor K-group $K_{2}^{M} (K)$ , which is by definition

K^× ⊗ K^× / (a ⊗ (1−a), a ∈ K^× \ {1})

By the first property it even factors over $K_{2}^{M} (K) / 2$ . This is the first step towards the Milnor conjecture.

Interpretation as an algebra

The Hilbert symbol can also be used to denote the central simple algebra over K with basis 1,i,j,k and multiplication rules $i^{2} = a$ , $j^{2} = b$ , $i j = - j i = k$ . In this case the algebra represents an element of order 2 in the Brauer group of K, which is identified with -1 if it is a division algebra and +1 if it is isomorphic to the algebra of 2 by 2 matrices.

Hilbert symbols over the rationals

For a place v of the rational number field and rational numbers a, b we let (a, b)_v denote the value of the Hilbert symbol in the corresponding completion Q_v. As usual, if v is the valuation attached to a prime number p then the corresponding completion is the p-adic field and if v is the infinite place then the completion is the real number field.

Over the reals, (a, b)_∞ is +1 if at least one of a or b is positive, and −1 if both are negative.

Over the p-adics with p odd, writing $a = p^{α} u$ and $b = p^{β} v$ , where u and v are integers coprime to p, we have

(a, b)_{p} = (- 1)^{α β ϵ (p)} {(\frac{u}{p})}^{β} {(\frac{v}{p})}^{α}

, where

ϵ (p) = (p - 1) / 2

and the expression involves two Legendre symbols.

Over the 2-adics, again writing $a = 2^{α} u$ and $b = 2^{β} v$ , where u and v are odd numbers, we have

(a, b)_{2} = (- 1)^{ϵ (u) ϵ (v) + α ω (v) + β ω (u)}

, where

ω (x) = (x^{2} - 1) / 8 .

It is known that if v ranges over all places, (a, b)_v is 1 for almost all places. Therefore the following product formula

\prod_{v} (a, b)_{v} = 1

makes sense. It is equivalent to the law of quadratic reciprocity.

Kaplansky radical

The Hilbert symbol on a field F defines a map

(\cdot, \cdot) : F^{*} / F^{* 2} \times F^{*} / F^{* 2} \to B r (F)

where Br(F) is the Brauer group of F. The kernel of this mapping, the elements a such that (a,b)=1 for all b, is the Kaplansky radical of F.^[1]

The radical is a subgroup of F^*/F^*2, identified with a subgroup of F^*. The radical contains is equal to F^* if and only if F is not formally real and has u-invariant at most 2.^[2] In the opposite direction, a field with radical F^*2 is termed a Hilbert field.^[3]

The general Hilbert symbol

If K is a local field containing the group of nth roots of unity for some positive integer n prime to the characteristic of K, then the Hilbert symbol (,) is a function from K*×K* to μ_n. In terms of the Artin symbol it can be defined by^[4]

(a, b) \sqrt[n]{b} = (a, K (\sqrt[n]{b} / K)) \sqrt[n]{b}

Hilbert originally defined the Hilbert symbol before the Artin symbol was discovered, and his definition (for n prime) used the power residue symbol when K has residue characteristic coprime to n, and was rather complicated when K has residue characteristic dividing n.

Properties

The Hilbert symbol is (multiplicatively) bilinear:

(ab,c) = (a,c)(b,c)

(a,bc) = (a,b)(a,c)

skew symmetric:

(a,b) = (b,a)^–1

nondegenerate:

(a,b)=1 for all b if and only if a is in K*ⁿ

It detects norms (hence the name norm residue symbol):

(a,b)=1 if and only if a is a norm of an element in K(ⁿ√b)

It has the "symbol" properties:

(a,1–a)=1, (a,–a)=1.

Hilbert's reciprocity law

Hilbert's reciprocity law states that if a and b are in an algebraic number field containing the nth roots of unity then^[5]

\prod_{p} (a, b)_{p} = 1

where the product is over the finite and infinite primes p of the number field, and where (,)_{p is the Hilbert symbol of the completion at p. Hilbert's reciprocity law follows from the Artin reciprocity law and the definition of the Hilbert symbol in terms of the Artin symbol.}

Power residue symbol

If K is a number field containing the nth roots of unity, p is a prime ideal not dividing n, π is a prime element of the local field of p, and a is coprime to p, then the power residue symbol (Template:Su) is related to the Hilbert symbol by^[6]

(\binom{a}{p}) = (π, a)_{p}

The power residue symbol is extended to fractional ideals by multiplicativity, and defined for elements of the number field by putting (Template:Su)=(Template:Su) where (b) is the principal ideal generated by b. Hilbert's reciprocity law then implies the following reciprocity law for the residue symbol, for a and b prime to each other and to n:

(\binom{a}{b}) = (\binom{b}{a}) \prod_{p | n, \infty} (a, b)_{p}

External links

Template:Eom
HilbertSymbol at Mathworld

References

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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
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.

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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

↑ Lam (2005) pp.450–451
↑ Lam (2005) p.451
↑ Lam (2005) p.455
↑ Neukirch (1999) p.333
↑ Neukirch (1999) p.334
↑ Neukirch (1999) p.336

[Lam4501-1] Lam (2005) pp.450–451

[Lam451-2] Lam (2005) p.451

[Lam455-3] Lam (2005) p.455

[Neu333-4] Neukirch (1999) p.333

[Neu334-5] Neukirch (1999) p.334

[Neu336-6] Neukirch (1999) p.336

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+In [[mathematics]], the '''Hilbert symbol''' or '''norm-residue symbol''' is a function (–, –) from ''K''<sup>&times;</sup> &times; ''K''<sup>&times;</sup> to the group of ''n''th roots of unity in a [[local field]] ''K'' such as the fields of [[real number|reals]] or [[p-adic number]]s . It is related to [[reciprocity law (mathematics)|reciprocity laws]], and can be defined in terms of the [[Artin symbol]] of [[local class field theory]]. The Hilbert symbol was introduced by {{harvs|txt|first=David|last=Hilbert|authorlink=David Hilbert|year=1897|loc=sections 64, 131|year2=1998|loc2=English translation}} in his [[Zahlbericht]], with the slight difference that he defined it for elements of global fields rather than for the larger local fields.
+The Hilbert symbol has been generalized to [[higher local field]]s.
+==Quadratic Hilbert symbol==
+Over a local field ''K'' whose [[multiplicative group]] of non-zero elements is ''K''<sup>&times;</sup>,
+the quadratic Hilbert symbol is the [[function (mathematics)|function]] (–, –) from ''K''<sup>&times;</sup> &times; ''K''<sup>&times;</sup> to {&minus;1,1} defined by
+:<math>(a,b)=\begin{cases}1,&\mbox{ if }z^2=ax^2+by^2\mbox{ has a non-zero solution }(x,y,z)\in K^3;\\-1,&\mbox{ if  not.}\end{cases}</math>
+===Properties===
+The following three properties follow directly from the definition, by choosing suitable solutions of the diophantine equation above:
+*If ''a'' is a square, then (''a'', ''b'') = 1 for all ''b''.
+*For all ''a'',''b'' in ''K''<sup>&times;</sup>,  (''a'', ''b'') = (''b'', ''a'').
+*For any ''a'' in ''K''<sup>&times;</sup> such that ''a''&minus;1 is also in ''K''<sup>&times;</sup>, we have (''a'', 1&minus;''a'') = 1.
+The (bi)multiplicativity, i.e.,
+:(''a'', ''b''<sub>1</sub>''b''<sub>2</sub>) = (''a'', ''b''<sub>1</sub>)&middot;(''a'', ''b''<sub>2</sub>)
+for any ''a'', ''b''<sub>1</sub> and ''b''<sub>2</sub> in ''K''<sup>&times;</sup> is, however, more difficult to prove, and requires the development of [[local class field theory]].
+The third property shows that the Hilbert symbol is an example of a [[Steinberg symbol]] and thus factors over the second [[Milnor K-theory|Milnor K-group]] <math>K^M_2 (K)</math>, which is by definition
+:''K''<sup>&times;</sup> ⊗ ''K''<sup>&times;</sup> / (''a'' ⊗ (1&minus;''a)'', ''a'' &isin; ''K''<sup>&times;</sup> \ {1})
+By the first property it even factors over <math>K^M_2 (K) / 2</math>. This is the first step towards the [[Milnor conjecture]].
+===Interpretation as an algebra===
+The Hilbert symbol can also be used to denote the [[central simple algebra]] over ''K'' with basis 1,''i'',''j'',''k'' and multiplication rules <math>i^2=a</math>, <math>j^2=b</math>, <math>ij=-ji=k</math>.  In this case the algebra represents an element of order 2 in the [[Brauer group]] of ''K'', which is identified with -1 if it is a division algebra and +1 if it is isomorphic to the algebra of 2 by 2 matrices.
+===Hilbert symbols over the rationals===
+For a [[valuation (algebra)|place]] ''v'' of the [[rational number field]] and rational numbers ''a'', ''b'' we let (''a'', ''b'')<sub>''v''</sub> denote the value of the Hilbert symbol in the corresponding [[complete space|completion]] '''Q'''<sub>''v''</sub>. As usual, if ''v'' is the valuation attached to a prime number ''p'' then the corresponding completion is the [[P-adic number|p-adic field]] and if ''v'' is the infinite place then the completion is the [[real number]] field.
+Over the reals, (''a'', ''b'')<sub>&infin;</sub> is +1 if at least one of ''a'' or ''b'' is positive, and &minus;1 if both are negative.
+Over the p-adics with ''p'' odd, writing <math>a = p^{\alpha} u</math> and <math>b = p^{\beta} v</math>, where ''u'' and ''v'' are integers [[coprime]] to ''p'', we have
+:<math>(a,b)_p = (-1)^{\alpha\beta\epsilon(p)} \left(\frac{u}{p}\right)^\beta \left(\frac{v}{p}\right)^\alpha</math>, where <math>\epsilon(p) = (p-1)/2</math>
+and the expression involves two [[Legendre symbol]]s.
+Over the 2-adics, again writing <math>a = 2^\alpha u</math> and <math>b = 2^\beta v</math>, where ''u'' and ''v'' are [[odd number]]s, we have
+:<math>(a,b)_2 = (-1)^{\epsilon(u)\epsilon(v) + \alpha\omega(v) + \beta\omega(u)}</math>, where <math>\omega(x) = (x^2-1)/8.</math>
+It is known that if ''v'' ranges over all places, (''a'', ''b'')<sub>''v''</sub> is 1 for almost all places. Therefore the following product formula
+:<math>\prod_v (a,b)_v = 1</math>
+makes sense. It is equivalent to the law of [[quadratic reciprocity]].
+===Kaplansky radical===
+The Hilbert symbol on a field ''F'' defines a map
+:<math> (\cdot,\cdot) : F^*/F^{*2} \times F^*/F^{*2} \rightarrow \mathop{Br}(F) </math>
+where Br(''F'') is the Brauer group of ''F''.  The kernel of this mapping, the elements ''a'' such that (''a'',''b'')=1 for all ''b'', is the '''Kaplansky radical''' of ''F''.<ref name=Lam4501>Lam (2005) pp.450–451</ref>
+The radical is a subgroup of F<sup>*</sup>/F<sup>*2</sup>, identified with a subgroup of F<sup>*</sup>.  The radical contains is equal to F<sup>*</sup> if and only if ''F'' is not [[formally real]] and has [[u-invariant|''u''-invariant]] at most 2.<ref name=Lam451>Lam (2005) p.451</ref>  In the opposite direction, a field with radical F<sup>*2</sup> is termed a '''Hilbert field'''.<ref name=Lam455>Lam (2005) p.455</ref>
+==The general Hilbert symbol==
+If ''K'' is a local field containing the group of ''n''th roots of unity for some positive integer ''n'' prime to the characteristic of ''K'', then the Hilbert symbol (,) is a function from ''K''*×''K''* to μ<sub>''n''</sub>.  In terms of the Artin symbol it can be defined by<ref name=Neu333>Neukirch (1999) p.333</ref>
+:<math> (a,b)\sqrt[n]{b} = (a,K(\sqrt[n]{b}/K))\sqrt[n]{b}</math>
+Hilbert originally defined the Hilbert symbol before the Artin symbol was discovered, and his definition (for ''n'' prime) used the power residue symbol when ''K'' has residue characteristic coprime to ''n'', and was rather complicated when ''K'' has residue characteristic dividing ''n''.
+===Properties===
+The Hilbert symbol is (multiplicatively) bilinear:
+:(''ab'',''c'') = (''a'',''c'')(''b'',''c'')
+:(''a'',''bc'') = (''a'',''b'')(''a'',''c'')
+skew symmetric:
+:(''a'',''b'') = (''b'',''a'')<sup>–1</sup>
+nondegenerate:
+: (''a'',''b'')=1 for all ''b'' if and only if ''a'' is in ''K''*<sup>''n''</sup>
+It detects norms (hence the name norm residue symbol):
+:(''a'',''b'')=1 if and only if ''a'' is a norm of an element in ''K''(<sup>''n''</sup>&radic;''b'')
+It has the [[Steinberg symbol|"symbol" properties]]:
+:(''a'',1–''a'')=1, (''a'',–a)=1.
+===Hilbert's reciprocity law===
+Hilbert's reciprocity law states that if ''a'' and ''b'' are in an algebraic number field containing the ''n''th roots of unity then<ref name=Neu334>Neukirch (1999) p.334</ref>
+:<math>\prod_p (a,b)_p=1</math>
+where the product is over the finite and infinite primes ''p'' of the number field, and where (,)<sub>''p''</sup> is the Hilbert symbol of the completion at ''p''. Hilbert's reciprocity law follows from the [[Artin reciprocity law]] and the definition of the Hilbert symbol in terms of the Artin symbol.
+===Power residue symbol===
+If ''K'' is a number field containing the ''n''th roots of unity, ''p'' is a prime ideal not dividing ''n'', π is a prime element of the local field of ''p'', and ''a'' is coprime to ''p'', then the [[power residue symbol]] ({{su|p=''a''|b=''p''}}) is related to the Hilbert symbol by<ref name=Neu336>Neukirch (1999) p.336</ref>
+:<math>\binom{a}{p} = (\pi,a)_p</math>
+The power residue symbol is extended to fractional ideals by multiplicativity, and defined for elements of the number field
+by putting ({{su|p=''a''|b=''b''}})=({{su|p= ''a''|b=(''b'')}}) where (''b'') is the principal ideal generated by ''b''.
+Hilbert's reciprocity law then implies the following reciprocity law for the residue symbol, for ''a'' and ''b'' prime to each other and to ''n'':
+:<math>\binom{a}{b}=\binom{b}{a}\prod_{p|n,\infty}(a,b)_p</math>
+==External links==
+*{{eom|id=Norm-residue_symbol|title=Norm-residue symbol}}
+*[http://mathworld.wolfram.com/HilbertSymbol.html HilbertSymbol] at [[Mathworld]]
+==References==
+{{reflist}}
+*{{Citation | first1=Z. I. | last1=Borevich | author1-link=Zenon Ivanovich Borevich | first2=I. R. | last2=Shafarevich | author2-link=Igor Shafarevich | title = Number theory | publisher = Academic Press | year = 1966 | isbn=0-12-117851-X | zbl=0145.04902 }}
+*{{Citation | last1=Hilbert | first1=David | author1-link=David Hilbert | title=Die Theorie der algebraischen Zahlkörper | url=http://resolver.sub.uni-goettingen.de/purl?GDZPPN002115344 | language=German | year=1897 | journal=[[Jahresbericht der Deutschen Mathematiker-Vereinigung]] | issn=0012-0456 | volume=4 | pages=175–546 }}
+*{{Citation | last1=Hilbert | first1=David | author1-link=David Hilbert | title=The theory of algebraic number fields | url=http://books.google.com/books?id=_Q2h83Bm94cC | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-3-540-62779-1 | id={{MR|1646901}} | year=1998}}
+* {{citation | title=Introduction to Quadratic Forms over Fields | volume=67 | series=Graduate Studies in Mathematics | first=Tsit-Yuen | last=Lam | authorlink=Tsit Yuen Lam | publisher=[[American Mathematical Society]] | year=2005 | isbn=0-8218-1095-2 | zbl=1068.11023 }}
+* {{Citation | last1=Milnor | first1=John Willard | author1-link=John Milnor | title=Introduction to algebraic ''K''-theory | publisher=[[Princeton University Press]] | series=Annals of Mathematics Studies | mr = 0349811 | year=1971 | volume=72 | zbl=0237.18005 }}
+* {{citation | last=Neukirch | first=Jürgen | author-link=Jürgen Neukirch | title=Algebraic number theory | others=Translated from the German by Norbert Schappacher | series=Grundlehren der Mathematischen Wissenschaften | volume=322 | location=Berlin | publisher=[[Springer-Verlag]] | year=1999 | isbn=3-540-65399-6 | zbl=0956.11021 }}
+* {{Citation | last1=Serre | first1=Jean-Pierre | author1-link=Jean-Pierre Serre | title=A Course in Arithmetic | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=[[Graduate Texts in Mathematics]] | isbn=978-3-540-90040-5 | year=1996 | volume=7 | zbl=0256.12001 }}
+* {{Citation | last1=Vostokov | first1=S. V. | last2=Fesenko | first2=I. B. | title=Local fields and their extensions | url=http://www.maths.nott.ac.uk/personal/ibf/book/book.html | series=Translations of Mathematical Monographs | volume=121 | publisher=[[American Mathematical Society]] | location=Providence, R.I. | isbn=978-0-8218-3259-2 | year=2002 | zbl=1156.11046 }}
+[[Category:Class field theory]]
+[[Category:Quadratic forms]]

Lindley's paradox: Difference between revisions

Revision as of 23:14, 21 December 2013

Contents

Quadratic Hilbert symbol

Properties

Interpretation as an algebra

Hilbert symbols over the rationals

Kaplansky radical

The general Hilbert symbol

Properties

Hilbert's reciprocity law

Power residue symbol

External links

References

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Lindley's paradox: Difference between revisions

Revision as of 23:14, 21 December 2013

Quadratic Hilbert symbol

Properties

Interpretation as an algebra

Hilbert symbols over the rationals

Kaplansky radical

The general Hilbert symbol

Properties

Hilbert's reciprocity law

Power residue symbol

External links

References

Navigation menu

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