Reduced ramification polynomial ($g_{\pi }(X)=f(\pi (1+X))/({\pi }^{n}X)$)

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.

In a totally ramified Galois extension, the uniformizer criterion for lower ramification groups gives $\sigma \in G_m\setminus G_{m+1}$ exactly when $v_L((\sigma (\pi )-\pi )/\pi )=m$. The Newton polygon root valuation theorem therefore identifies a slope $-m$ in the coefficient-index convention with a lower ramification jump at $m$. Its horizontal length is the number of automorphisms in that difference. Under the reflected Newton polygon convention the same jump has slope $+m$.