= Quadratic extensions of an odd-residue-characteristic local field
{title2=$K^\times/K^{\times2}\cong(\mathbb Z/2\mathbb Z)^2$}
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.
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