Square-class group of a field ($K^{\times }/(K^{\times }{)}^2$)

The square-class group identifies two nonzero elements of a field when their quotient is a square. Multiplication makes it an abelian group of exponent two, hence a vector space over ${\mathbb{F}}_2$. It records quadratic extensions and the images of Kummer theory for $n=2$.

The valuation contributes one factor $C_2$. The unit decomposition of the 2-adic Gaussian field contributes ${\mu }_4/{\mu }_4^2\cong C_2$ and the additive quotient ${\mathbb{Z}}_2[i]/2{\mathbb{Z}}_2[i]\cong C_2^2$. Thus there are sixteen square classes and fifteen quadratic extensions over this field.

The square class of $a\in K^{\ast }$ is its image in the square-class group of a field $K^{\ast }/K^{\ast 2}$. Two nonzero elements have the same square class exactly when their quotient is a square. Every element of this quotient has order dividing $2$. Over the rational numbers, a square class has a unique signed squarefree integer representative.

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.