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| In [[mathematics]], the '''Hilbert symbol''' or '''norm-residue symbol''' is a function (–, –) from ''K''<sup>×</sup> × ''K''<sup>×</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.
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| The Hilbert symbol has been generalized to [[higher local field]]s.
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| ==Quadratic Hilbert symbol==
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| Over a local field ''K'' whose [[multiplicative group]] of non-zero elements is ''K''<sup>×</sup>,
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| the quadratic Hilbert symbol is the [[function (mathematics)|function]] (–, –) from ''K''<sup>×</sup> × ''K''<sup>×</sup> to {−1,1} defined by
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| :<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>
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| ===Properties===
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| The following three properties follow directly from the definition, by choosing suitable solutions of the diophantine equation above:
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| *If ''a'' is a square, then (''a'', ''b'') = 1 for all ''b''.
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| *For all ''a'',''b'' in ''K''<sup>×</sup>, (''a'', ''b'') = (''b'', ''a'').
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| *For any ''a'' in ''K''<sup>×</sup> such that ''a''−1 is also in ''K''<sup>×</sup>, we have (''a'', 1−''a'') = 1.
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| The (bi)multiplicativity, i.e.,
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| :(''a'', ''b''<sub>1</sub>''b''<sub>2</sub>) = (''a'', ''b''<sub>1</sub>)·(''a'', ''b''<sub>2</sub>)
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| for any ''a'', ''b''<sub>1</sub> and ''b''<sub>2</sub> in ''K''<sup>×</sup> is, however, more difficult to prove, and requires the development of [[local class field theory]].
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| 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
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| :''K''<sup>×</sup> ⊗ ''K''<sup>×</sup> / (''a'' ⊗ (1−''a)'', ''a'' ∈ ''K''<sup>×</sup> \ {1})
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| By the first property it even factors over <math>K^M_2 (K) / 2</math>. This is the first step towards the [[Milnor conjecture]].
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| ===Interpretation as an algebra===
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| 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.
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| ===Hilbert symbols over the rationals===
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| 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.
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| Over the reals, (''a'', ''b'')<sub>∞</sub> is +1 if at least one of ''a'' or ''b'' is positive, and −1 if both are negative.
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| 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
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| :<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>
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| and the expression involves two [[Legendre symbol]]s.
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| 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
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| :<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>
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| 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
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| :<math>\prod_v (a,b)_v = 1</math>
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| makes sense. It is equivalent to the law of [[quadratic reciprocity]].
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| ===Kaplansky radical===
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| The Hilbert symbol on a field ''F'' defines a map
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| :<math> (\cdot,\cdot) : F^*/F^{*2} \times F^*/F^{*2} \rightarrow \mathop{Br}(F) </math>
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| 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>
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| 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>
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| ==The general Hilbert symbol==
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| 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>
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| :<math> (a,b)\sqrt[n]{b} = (a,K(\sqrt[n]{b}/K))\sqrt[n]{b}</math>
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| 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''.
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| ===Properties===
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| The Hilbert symbol is (multiplicatively) bilinear:
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| :(''ab'',''c'') = (''a'',''c'')(''b'',''c'')
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| :(''a'',''bc'') = (''a'',''b'')(''a'',''c'')
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| skew symmetric:
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| :(''a'',''b'') = (''b'',''a'')<sup>–1</sup>
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| nondegenerate:
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| : (''a'',''b'')=1 for all ''b'' if and only if ''a'' is in ''K''*<sup>''n''</sup>
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| It detects norms (hence the name norm residue symbol):
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| :(''a'',''b'')=1 if and only if ''a'' is a norm of an element in ''K''(<sup>''n''</sup>√''b'')
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| It has the [[Steinberg symbol|"symbol" properties]]:
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| :(''a'',1–''a'')=1, (''a'',–a)=1.
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| ===Hilbert's reciprocity law===
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| 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>
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| :<math>\prod_p (a,b)_p=1</math>
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| 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.
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| ===Power residue symbol===
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| 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>
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| The power residue symbol is extended to fractional ideals by multiplicativity, and defined for elements of the number field
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| by putting ({{su|p=''a''|b=''b''}})=({{su|p= ''a''|b=(''b'')}}) where (''b'') is the principal ideal generated by ''b''.
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| 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'':
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| :<math>\binom{a}{b}=\binom{b}{a}\prod_{p|n,\infty}(a,b)_p</math>
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| ==External links==
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| *{{eom|id=Norm-residue_symbol|title=Norm-residue symbol}}
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| *[http://mathworld.wolfram.com/HilbertSymbol.html HilbertSymbol] at [[Mathworld]]
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| ==References==
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| {{reflist}}
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| *{{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 }}
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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}}
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| * {{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 }}
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| * {{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 }}
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| * {{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 }}
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| * {{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 }}
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| * {{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 }}
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| [[Category:Class field theory]]
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| [[Category:Quadratic forms]]
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