 Quadratic extensions of the 2-adic numbers

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 ${\mathbb{Q}}_2(\sqrt{5})$: $(1+\sqrt{5})/2$ 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.