= Hilbert-symbol annihilators of units and unramified square classes
{c}
{title2=$\Delta_{\rm units}^{\perp}=\Delta_{\rm unr},\quad\Delta_{\rm unr}^{\perp}=\Delta_{\rm units}$}
In the <square-class group of a p-adic field>, the unit <subgroup> consists of classes of even <valuation>. The unramified <subgroup> consists of the trivial class and the unique nonsquare class defining an <unramified extension> of degree two. For odd $p$, the two <subgroups> coincide with $\{1,u\}$ for a nonsquare unit $u$. For $p=2$, they are $\{1,-1,5,-5\}$ and $\{1,5\}$. Norms from an unramified quadratic <local field> extension have even <valuation>, and every unit is a norm. The norm characterization of the <quadratic Hilbert symbol> therefore gives both displayed annihilator identities. Over $\mathbb Q_2$, the unramified class is $5$ because $(1+\sqrt5)/2$ satisfies $T^2-T-1$, whose reduction modulo two is irreducible and separable.
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