The Divisors of $x^{2m} + x$ of Constant Derivatives and Degree $2^{m - 2}$

Abstract

This paper provides a new proof of the fact that the polynomials of degree $2^{m-2}$ over the Galois field $GF(2^m) (m \geq 2)$, which are fully reducible and admit no multiple factor (i.e., which divide x^{2^m} + x$) and whose derivatives are constant are affine polynomials. The author determines explicitly these polynomials, which are related to a problem in coding theory that is recounted.