PlanetMath: Hilbert symbol

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Hilbert symbol

(Definition)

Let be any local field. For any two nonzero elements $a,b \in K^\times$ , we define:

$\begin{displaymath} (a,b) := \begin{cases} +1 & \text{ if $z^2 = ax^2 + by^2$ ha... ...neq (0,0,0)$ in $K^3$,} \ -1 & \text{ otherwise.} \end{cases}\end{displaymath}$

The number

is called the Hilbert symbol of

and

"Hilbert symbol" is owned by djao.

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Cross-references: number, local field
There is 1 reference to this entry.

This is version 2 of Hilbert symbol, born on 2002-07-21, modified 2002-07-21.
Object id is 3175, canonical name is HilbertSymbol.
Accessed 2299 times total.

Classification:

AMS MSC:	11S31 (Number theory :: Algebraic number theory: local and $p$-adic fields :: Class field theory; $p$-adic formal groups)
	11S80 (Number theory :: Algebraic number theory: local and $p$-adic fields :: Other analytic theory )

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