The generalized Eisenstein criterion (Q2639911)

The author proves the following slight generalization of the Eisenstein criterion: Let F be a local field with normalized discrete valuation \(\nu\). Let \(f(x)=x^ n+a_ 1x^{n-1}+...+a_ n\in F[x]\) be such that \(\nu (a_ n)=d\geq 1\) and \(\nu (a_ i)\geq d\), where d is relatively prime to n. Then f(x) is irreducible in F[x] and a zero \(\alpha\) of f(x) generates a totally ramified extension F(\(\alpha\))/F. As an application, he shows that certain tamely ramified extensions of local fields are cyclic or metacyclic.