= Explicit quadratic Hilbert symbols over the p-adic numbers
{title2=$(p^\alpha u,p^\beta v)_p$}
For odd $p$ and <p-adic units> $u,v$, the <quadratic Hilbert symbol> is
$$
(p^\alpha u,p^\beta v)_p=(-1)^{\alpha\beta(p-1)/2}\left(\frac up\right)^\beta\left(\frac vp\right)^\alpha.
$$
The <Legendre symbols> record the two unit <square classes>. For $p=2$, put $\epsilon(u)=(u-1)/2\bmod2$ and $\omega(u)=(u^2-1)/8\bmod2$. Then
$$
(2^\alpha u,2^\beta v)_2=(-1)^{\epsilon(u)\epsilon(v)+\alpha\omega(v)+\beta\omega(u)}.
$$
The basis $-1,2,5$ of the <square-class group of the 2-adic numbers> gives this formula: its only negative basis pairings are $(-1,-1)_2$ and $(2,5)_2=(5,2)_2$. The first is obstructed by sums of squares modulo four, the second by $x^2-2y^2$ modulo eight; the remaining pairings have explicit <field norm> representatives.
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