= Quadratic branch parity criterion
{title2=$e_p=2\ \Longleftrightarrow\ v_p(a)\text{ is odd}$}
In <characteristic zero> with algebraically closed constants, a local unit has a square root. Thus $K(T)(\sqrt a)$ near a point is equivalent to adjoining the square root of the local parameter to the power $v_p(a)$. Even order is unramified; odd order gives <ramification index> two. For a <polynomial> $a$, its order at infinity is minus its degree, so infinity branches exactly for odd degree.
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