Quadratic extensions from square classes ($\{K(\sqrt{a})\}/K\text{-isomorphism}⟷K^{\times }/K^{\times 2}\setminus \{1\}$)

For a field of characteristic different from two, adjoining a square root of a nonsquare gives every quadratic extension. If two such extensions are isomorphic over $K$, write $\sqrt{a}=x+y\sqrt{b}$; the vanishing of its cross coefficient forces $x=0$, giving $a/b=y^2$. This explains why one subtracts the identity class when counting extensions from the square-class group.