Roots of Affine Polynomials

Abstract

A Galois field F = GF(q) of order q = p h is considered in the chapter, where p is a prime, and K = GF(q n ) is considered an algebraic extension of a given degree n > 1. An affine polynomial—of K[x]over F—is a polynomial that can be expressed as deference between L(x) and b, where b belongs to K. Hence, the determination of (eventual) roots of the polynomial in K can be reduced to the determination of solutions of a linear system of equations in indeterminates x i and with coefficients in F. This procedure is, however, tedious also in the most simply cases and does not decise a priori how many roots in K exist. The conditional equation for affine polynomial has roots in K only if the certain system of linear equations has solutions. The roots of the equation are expressible as functions of certain solutions of the linear system. In particular, the obtained results are useful also in the case where the coefficients of the affine polynomial are not constant.