Swan-like reducibility for Type I pentanomials over a binary field

TL;DR

This paper extends Swan's classical results to analyze the reducibility of a specific class of pentanomials over binary fields, providing a new criterion for their factorization based on discriminant calculations.

Contribution

It determines the parity of the number of irreducible factors for Type I pentanomials over F2 with even n, using Stickelberger-Swan theorem and Newton's formula.

Findings

Parity of irreducible factors for Type I pentanomials determined

New reducibility criterion based on discriminant analysis

Extension of Swan's classical results to pentanomials

Abstract

Swan (Pacific J. Math. 12(3) (1962), 1099-1106) characterized the parity of the number of irreducible factors of trinomials over $F_2$. Many researchers have recently obtained Swan-like results on determining the reducibility of polynomials over finite fields. In this paper, we determine the parity of the number of irreducible factors for so-called Type I pentanomial $f(x)=x^m+x^{n+1}+x^n+x+1$ over $F_2$ with even $n$. Our result is based on the Stickelberger-Swan theorem and Newton's formula which is very useful for the computation of the discriminant of a polynomial.