Hilbert symbol

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In mathematics, given a local field K, whose multiplicative group of non-zero elements is K*, the Hilbert symbol is an algebraic construction, extracted from reciprocity laws, and important in the formulation of local class field theory. As the name suggests, it was in some sense introduced by David Hilbert, although it would be anachronistic to say that of the local field formulation.

Explicitly, it is the function (.,.) from K* x K* to {−1,1} defined by

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

1 Properties
2 Interpretation as an algebra
3 Hilbert symbols over the rationals
4 External links
5 References

[edit] Properties

If a is a square, then (a,b) = 1 for all b.
For any $a, b 1, b 2$ in K*, we have $(a, b 1 b 2) = (a, b 1)(a, b 2)$ .
For all a,b in K*, $(a,, b) = (b, a)$ .
For any a in K*, we have $(a,1 - a) = 1$ .

[edit] 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 quaternion algebra and +1 if it is a matrix algebra.

[edit] Hilbert symbols over the rationals

For a valuation 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 valuation then the completion is the real number field.

Over the reals, $(a,b)_\infty$ 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

$(a,b)_p = (-1)^{\alpha\beta\epsilon(p)} \left(\frac{a}{p}\right)^\beta \left(\frac{b}{p}\right)^\alpha$ , 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) + αω(v) + βω(u)

, where

ω(x) = (x 2 - 1) / 8

The product formula

∏	(a,b)_v = 1
v

where the product is taken over all valuations v is equivalent to the law of quadratic reciprocity.

[edit] External links

Steinberg symbol at the Encyclopaedia of Mathematics
HilbertSymbol at Mathworld

[edit] References

Z. I. Borevich, I. R. Shafarevich (1966). Number theory. Academic Press. ISBN 0-12-117851-X.
J. Milnor, Introduction to algebraic K-theory , Ann. of Math. Stud. 72, Princeton Univ. Press (1971) ISBN 0-691-08101-8
Local fields and their extensions by I. B. Fesenko and S. V. Vostokov ISBN 978-0-8218-3259-2
J.-P. Serre, A course in arithmetic, Graduate Texts in Mathematics 7, Springer Verlag (1973) ISBN 3-540-90040-3