On binomials and algebraic closure of some pseudofinite fields

TL;DR

This paper provides a criterion for the irreducibility of certain polynomials over pseudofinite fields and describes the algebraic closure of specific pseudofinite fields of zero characteristic.

Contribution

It introduces a new criterion for polynomial irreducibility and explicitly characterizes the algebraic closure of some pseudofinite fields of zero characteristic.

Findings

Criterion for irreducibility of x^n - g over pseudofinite fields

Explicit description of algebraic closure of certain pseudofinite fields

Applications to fields of zero characteristic

Abstract

We give a criterion when a polynomial $x^n-g$ is irreducible over a pseudofinite field. As an application we give an explicit description of algebraic closure of some pseudofinite fields of zero characteristic.