 Square-class group of a p-adic field

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.