On irreducibility of certain Schur polynomials over fields of finite characteristic

TL;DR

This paper provides an elementary proof that certain Schur polynomials are irreducible over fields of finite characteristic under specific conditions on their exponents, expanding understanding of polynomial irreducibility in algebra.

Contribution

The paper introduces a new elementary proof establishing irreducibility of specific Schur polynomials over finite characteristic fields, under particular conditions on the exponents.

Findings

Schur polynomials with specified exponent conditions are irreducible over fields of characteristic p.

Irreducibility holds when differences of exponents are greater than 1, not divisible by p, and consecutive differences are coprime.

Provides a simplified proof approach for polynomial irreducibility in finite characteristic settings.

Abstract

We present an elementary proof that the Schur polynomial corresponding to an increasing sequence of exponents (c_0,..., c_{n-1}) with c_0 = 0 is irreducible over every field of characteristic p whenever the numbers d_i = c_{i+1} - c_i are all greater than 1, not divisible by p, and satisfy gcd(d_i, d_{i+1}) = 1 for every i.