= Square-class group of a p-adic field
{title2=$\mathbb Q_p^\times/(\mathbb Q_p^\times)^2$}
For a finite extension $K/\mathbb Q_p$, its <square-class group of a field> is $K^\times/(K^\times)^2$. When $p$ is odd, parity of the <valuation> and the square class of the unit residue give $C_2\times C_2$. Representatives are $1,u,\pi,u\pi$, where $\pi$ is a <uniformizer> of $K$ and $u$ has nonsquare residue. The <Hensel lemma> lifts square roots of units with square residue.
For $K=\mathbb Q_2$, the group is $C_2^3$, with generators $-1,5,2$; a unit is a square exactly when it is one modulo eight. More generally, if $K/\mathbb Q_2$ has degree $d$, the group has order $2^{d+2}$. Indeed, the <principal-unit group of a mixed-characteristic local field> has a finite cyclic 2-power torsion factor and a free $\mathbb Z_2$ factor of rank $d$, each contributing its quotient by squares, and the valuation contributes one further factor $C_2$. The specific representatives $1,u,p,up$ therefore apply to $\mathbb Q_p$ for odd $p$; a ramified extension needs its own uniformizer.
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