Hilbert symbol

The Hilbert symbol (after David Hilbert ) is a short form that is used in algebraic number theory . For a local field with the multiplicative group it is defined as the following figure: ${\ displaystyle K}$ ${\ displaystyle K ^ {*}}$

{\ displaystyle {\ begin {aligned} K ^ {*} \ times K ^ {*} & \ rightarrow \ {- 1,1 \} \\ (a, b) & \ mapsto {\ begin {cases} 1, & {\ text {if}} \ z ^ {2} = ax ^ {2} + by ^ {2} \ {\ text {one not}} {\ mbox {-}} {\ text {trivial solution} } \ (x, y, z) \ in K ^ {3} \ {\ text {owns}}; \\ - 1, & {\ text {otherwise}} \ end {cases}} \ end {aligned}} }

A solution is called trivial if the following applies. ${\ displaystyle x = y = z = 0}$

properties

An element in is a square if and only if applies to all . ${\ displaystyle a}$ ${\ displaystyle K ^ {*}}$ ${\ displaystyle (a, b) = 1}$ ${\ displaystyle b \ in K ^ {*}}$
For those in the following applies: . ${\ displaystyle a, b}$ ${\ displaystyle K ^ {*}}$ ${\ displaystyle (a, b) = (b, a)}$
For those in the following applies: . ${\ displaystyle a, b_ {1}, b_ {2}}$ ${\ displaystyle K ^ {*}}$ ${\ displaystyle (a, b_ {1} b_ {2}) = (a, b_ {1}) (a, b_ {2})}$
For all in with valid . ${\ displaystyle a}$ ${\ displaystyle K ^ {*}}$ ${\ displaystyle a \ neq 1}$ ${\ displaystyle (a, 1-a) = 1}$

Jean-Pierre Serre : A course in arithmetic (= Graduate Texts in Mathematics. Vol. 7). Springer, New York NY et al. 1973, ISBN 3-540-90040-3 .