Quadratic extensions of an odd-residue-characteristic local field ($K^{\times }/K^{\times 2}\cong (\mathbb{Z}/2\mathbb{Z}{)}^2$)

For a p-adic field with odd residue characteristic, the square map is an isomorphism on $1+{\mathfrak{m}}_K$ by Hensel lemma. The parity of the valuation and the square class of the residue unit therefore determine the square-class group of a field. Its three nontrivial classes give the three quadratic extensions: one unramified extension and two totally ramified extensions.