Eisenstein polynomial

For a discrete valuation ring with uniformizer $\pi$, an Eisenstein polynomial is a monic polynomial whose nonleading coefficients are divisible by $\pi$ and whose constant coefficient is not divisible by ${\pi }^2$. It is irreducible, and adjoining one of its roots gives a totally ramified extension.

A polynomial may fail the Eisenstein polynomial test while a translate $f(T+a)$ passes it. Translation does not change the generated extension: if $f(\alpha )=0$, then $f(T+a)$ vanishes at $\alpha -a$. For $f(T)=T^3-m$, the translate by $m$ is Eisenstein at $3$ whenever $3\nmid m$ and $m\not\equiv \pm 1(\mathrm{mod}9)$.

The ramification polynomial records the normalised differences of conjugates of a uniformizer, including the zero difference from the identity automorphism. Its quotient by $X$ is the reduced ramification polynomial. In the usual coefficient-index convention, the finite slopes of its lower Newton polygon are negatives of the valuations of its nonzero roots; the coefficient at the zero root is treated as having infinite valuation.

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 $+m$.