Remarks on Eisenstein

TL;DR

This paper provides an explicit quantitative version of Eisenstein's theorem on algebraic power series, extending it to ramified and Laurent series, with applications including discriminant estimation of related number fields.

Contribution

It introduces a fully explicit form of Eisenstein's theorem suitable for applications and extends the theorem to ramified and Laurent series.

Findings

Explicit bounds for algebraic power series coefficients

Extension of Eisenstein's theorem to ramified and Laurent series

Application: estimation of discriminants of number fields

Abstract

We obtain a fully explicit quantitative version of the Eisenstein theorem on algebraic power series which is more suitable for certain applications than the existing version due to Dwork, Robba, Schmidt and van der Poorten. We also treat ramified series and Laurent series, and we demonstrate some applications; for instance, we estimate the discriminant of the number field generated by the coefficients.