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In [[abstract algebra]], '''Hilbert's Theorem 90''' (or '''Satz 90''') refers to an important result on [[cyclic extension]]s of [[field (mathematics)|field]]s (or to one of its generalizations) that leads to [[Kummer theory]]. In its most basic form, it states that if ''L''/''K'' is a cyclic extension of fields with Galois group ''G'' =''Gal''(''L''/''K'')
generated by an element '''s''' and if a is an element of ''L'' of [[relative norm]] 1, then there exists b in ''L'' such that


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:a = '''s'''(b)/b.
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The theorem takes its name from the fact that it is the 90th theorem in [[David Hilbert]]'s famous [[Zahlbericht]] {{harvs|last=Hilbert|year=1897|year2=1998}}, although it is originally due to {{harvs|txt|authorlink=Ernst Kummer|last=Kummer|year1=1855|loc1=p.213|year2=1861}}. Often a more general theorem due to {{harvs|txt|authorlink=Emmy Noether|first=Emmy|last=Noether|year=1933}} is given the name, stating that if ''L''/''K'' is a finite Galois extension of fields with Galois group ''G'' =''Gal''(''L''/''K''), then the first cohomology group is trivial:
:''H''<sup>1</sup>(''G'', ''L''<sup>×</sup>) = {1}


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Let ''L/K'' be the [[quadratic extension]] <math> \mathbb{Q}(i) / \mathbb{Q} </math>. The Galois group is cyclic of order 2, its generator s acting via conjugation:
<ul>
:<math> s:\, \, a+bi\mapsto a-bi\ . </math>
 
An element <math> x=a+bi</math> in ''L'' has norm <math> xx^{s}=a^2 +b^2</math>. An element of norm one corresponds to a rational solution of the equation <math>a^2+b^2=1</math> or in other words, a point with rational coordinates on the [[unit circle]]. Hilbert's Theorem 90 then states that every element ''y'' of norm one can be parametrized (with integral ''c,d'') as
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:<math> y={{c+di}\over{c-di}}={{c^2-d^2}\over{c^2+d^2}}+{2dc\over{c^2+d^2}}i
 
</math>
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which may be viewed as a rational parametrization of the rational points on the unit circle. Rational points <math>\, (x,y)=(a/c,b/c) </math> on the unit circle <math> x^2+y^2=1 </math> correspond to [[Pythagorean triple]]s, i.e. triples <math>\,(a,b,c)</math> of integers satisfying <math>\, a^2+b^2=c^2 </math>.
 
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The theorem can be stated in terms of [[group cohomology]]: if ''L''<sup>×</sup> is the [[multiplicative group]] of any (not necessarily finite) Galois extension ''L'' of a field ''K'' with corresponding Galois group ''G'', then


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:''H''<sup>1</sup>(''G'', ''L''<sup>×</sup>) = {1}.
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A further generalization using [[Non-abelian cohomology|non-abelian group cohomology]] states that if ''H'' is either the [[general linear group|general]] or [[special linear group]] over ''L'', then


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:''H''<sup>1</sup>(''G'',''H'') = {1}.
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This is a generalization since ''L''<sup>×</sup> = GL<sub>1</sub>(''L'').


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Another generalization is <math>H^1_{\acute{e}t}(X,\mathbf{G}_m) = H^1(X,\mathcal{O}_X^\times) = \mathrm{Pic}(X)</math> for ''X'' a scheme, and another one to [[Milnor K-theory]] plays a role in [[Vladimir Voevodsky|Voevodsky's]] proof of the [[Milnor conjecture]].
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*{{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 }}
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*{{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}}
<ul>
*{{Citation | last1=Kummer | first1=Ernst Eduard | title=Über eine besondere Art, aus complexen Einheiten gebildeter Ausdrücke. | url=http://resolver.sub.uni-goettingen.de/purl?GDZPPN002148927 | language=German | doi=10.1515/crll.1855.50.212 | year=1855 | journal=[[Journal für die reine und angewandte Mathematik]] | issn=0075-4102 | volume=50 | pages=212–232}}
 
*{{Citation | last1=Kummer | first1=Ernst Eduard | title=Zwei neue Beweise der allgemeinen Reciprocitätsgesetze unter den Resten und Nichtresten der Potenzen, deren Grad eine Primzahl ist | language=German | id=Reprinted in volume 1 of his collected works, pages 699–839 | year=1861 | journal=Abdruck aus den Abhandlungen der Kgl. Akademie der Wissenschaften zu Berlin}}
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*Chapter II of J.S. Milne, ''Class Field Theory'', available at his website [http://www.jmilne.org/math].
 
*{{Neukirch et al. CNF}}
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*{{Citation | last1=Noether | first1=Emmy | author1-link=Emmy Noether | title=Der Hauptgeschlechtssatz für relativ-galoissche Zahlkörper. | language=German | doi=10.1007/BF01452845 | id={{Zbl|0007.29501}} | year=1933 | journal=[[Mathematische Annalen]] | issn=0025-5831 | volume=108 | issue=1 | pages=411–419}}
 
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Revision as of 12:02, 31 December 2012

In abstract algebra, Hilbert's Theorem 90 (or Satz 90) refers to an important result on cyclic extensions of fields (or to one of its generalizations) that leads to Kummer theory. In its most basic form, it states that if L/K is a cyclic extension of fields with Galois group G =Gal(L/K) generated by an element s and if a is an element of L of relative norm 1, then there exists b in L such that

a = s(b)/b.

The theorem takes its name from the fact that it is the 90th theorem in David Hilbert's famous Zahlbericht Template:Harvs, although it is originally due to Template:Harvs. Often a more general theorem due to Template:Harvs is given the name, stating that if L/K is a finite Galois extension of fields with Galois group G =Gal(L/K), then the first cohomology group is trivial:

H1(G, L×) = {1}

Examples

Let L/K be the quadratic extension (i)/. The Galois group is cyclic of order 2, its generator s acting via conjugation:

s:a+biabi.

An element x=a+bi in L has norm xxs=a2+b2. An element of norm one corresponds to a rational solution of the equation a2+b2=1 or in other words, a point with rational coordinates on the unit circle. Hilbert's Theorem 90 then states that every element y of norm one can be parametrized (with integral c,d) as

y=c+dicdi=c2d2c2+d2+2dcc2+d2i

which may be viewed as a rational parametrization of the rational points on the unit circle. Rational points (x,y)=(a/c,b/c) on the unit circle x2+y2=1 correspond to Pythagorean triples, i.e. triples (a,b,c) of integers satisfying a2+b2=c2.

Cohomology

The theorem can be stated in terms of group cohomology: if L× is the multiplicative group of any (not necessarily finite) Galois extension L of a field K with corresponding Galois group G, then

H1(G, L×) = {1}.

A further generalization using non-abelian group cohomology states that if H is either the general or special linear group over L, then

H1(G,H) = {1}.

This is a generalization since L× = GL1(L).

Another generalization is He´t1(X,Gm)=H1(X,𝒪X×)=Pic(X) for X a scheme, and another one to Milnor K-theory plays a role in Voevodsky's proof of the Milnor conjecture.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  • Chapter II of J.S. Milne, Class Field Theory, available at his website [1].
  • Template:Neukirch et al. CNF
  • 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