Local Hilbert symbol

Let $F$ be a local field of characteristic different from two, and let $a,b\in F^\times$ . The Hilbert symbol $(a,b)_F\in\{+1,-1\}$ is $+1$ if the quaternion algebra generated by $i^2=a$ , $j^2=b$ , $ij=-ji$ is split, and $-1$ if it is a division algebra.

This page uses the sign-valued convention. The similar notation $(a,b)_F$ for the quaternion algebra itself denotes an algebra, not a sign.

Square classes

Multiplying either parameter by a nonzero square does not change the symbol. It therefore descends to $F^\times/F^{\times2}$ in each argument. The resulting pairing is symmetric and multiplicative in each variable.

Over $\mathbb R$ , the value is $-1$ exactly when both parameters are negative. Over $\mathbb C$ , it is always $+1$ .

The prime two matters

For odd integers $a,b$ ,

(a,b)_{\mathbb Q_2}=(-1)^{(a-1)(b-1)/4}.

Thus $(-1,-1)_{\mathbb Q_2}=-1$ , while $(-1,-1)_{\mathbb Q_p}=+1$ at every odd prime $p$ .

The sign is also distinct from the additive local Brauer invariant, whose quaternion values are $0$ and $\tfrac12$ modulo $\mathbb Z$ .

References

John Voight, Quaternion Algebras, author's text, Definition 12.4.1, Remark 12.4.2, Lemma 12.4.6, and formula (12.4.14).