Hilbert-symbol annihilators of units and unramified square classes (${\Delta }_{\mathrm{units}}^{\perp }={\Delta }_{\mathrm{unr}},{\Delta }_{\mathrm{unr}}^{\perp }={\Delta }_{\mathrm{units}}$)

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 $(1+\sqrt{5})/2$ satisfies $T^2-T-1$, whose reduction modulo two is irreducible and separable.