= Quadratic extensions of the 2-adic numbers
{title2=$\mathbb Q_2(\sqrt d),\quad d=-1,5,-5,2,-2,10,-10$}
The three generators $-1,2,5$ of the <square-class group of the 2-adic numbers> give these seven distinct extensions. In characteristic different from two, completing the square shows every quadratic extension is of this form, and comparing coefficients in $\sqrt e=a+b\sqrt d$ proves that two such fields coincide precisely when $e/d$ is a square. The only unramified member is $\mathbb Q_2(\sqrt5)$: $(1+\sqrt5)/2$ is integral with irreducible reduction $X^2+X+1$. Conversely any unramified quadratic extension has that residue field, and <Hensel lemma> lifts a residue root of $X^2-X-1$, recovering the same field.
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