= Reduced ramification polynomial
{title2=$g_\pi(X)=f(\pi(1+X))/(\pi^nX)$}
For a degree-$n$ totally ramified <Galois extension> and a <uniformizer> $\pi$, let $f$ be its <minimal polynomial>. Dividing the ramification <polynomial> $f(\pi(1+X))/\pi^n$ by its known factor $X$ gives the reduced ramification <polynomial>. It is monic of degree $n-1$, integral over the extension's <valuation ring>, and has roots $(\sigma(\pi)-\pi)/\pi$ for nonidentity automorphisms. Its constant coefficient is nonzero in a separable extension. Removing the zero root allows the usual <Newton polygon root valuation theorem> to apply directly.
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