= Quadratic extensions from square classes
{title2=$\{K(\sqrt a)\}/K\text{-isomorphism}\;\longleftrightarrow\;K^\times/K^{\times2}\setminus\{1\}$}
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>.
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