Polynomials and primitive elements in Galois rings

Kloosterman sums and primitive elements in Galois fields

1. Introduction. Let F q denote the finite (Galois) field of order q, a power of a prime p. The multiplicative group F

Factoring polynomials over Z4 and over certain Galois rings

It is known that univariate polynomials over finite local rings factor uniquely into primary pairwise coprime factors. Primary polynomials are not necessarily irreducible. Here we describe a factorisation into irreducible factors for primary polynomials over Z4 and more generally over Galois rings of characteristic p. An algorithm is also given. As an application, we factor x − 1 and x + 1 over...

Quasi-Primitive Rings and Density Theory

Galois Theory and Reducible Polynomials

This paper presents works on reducible polynomials in Galois theory. It gives a fundamental theorem about the Galois ideals associated with a product of groups. These results are used for the practical determination of Galois groups of polynomials of degrees up to 7. : : :). The following tools are used: resolvents (relative or absolute), p-adic methods, Grr obner bases or block systems. In gen...

Polynomials over Galois Field

( ) [ ] [ ] ( ) 1 2 0 0 1 0 0 1 2 0 1 1 0 2 n m n m a b a b a b x a b a b a b x a b x + = ⋅ + ⋅ + ⋅ + ⋅ + ⋅ + ⋅ + + ⋅ . The coefficient operations are performed using the operations for the field from which the coefficients were taken. • Such a collection of polynomials forms a commutative ring with identity. • Nonzero field elements are considered to be zero-degree polynomials. The zero elemen...