This is the material which I presented at the 60th birthday conference of my good friend José Luis Vicente in Seville in September 2001. It is based on the nine lectures, now called sections, which were given by me at Purdue in Spring 1997. This should provide a good calculational background for the Galois theory of vectorial (= additive) polynomials and their iterates.

Finding the approximate region containing all zeros of analytic polynomials is a well-studied problem. But number and regions complex-valued harmonic relatively fresh research area. It well known that trinomials are enclosed in some annular sectors take into account magnitude coefficients. Following Kennedy Dehmer, we provide zero inclusion general, particular, bound families certain region.

Permutation polynomials with few terms attracts researchers’ interest in recent years due to their simple algebraic form and some additional extraordinary properties. In this paper, by analyzing the quadratic factors of a fifth-degree polynomial and a seventh-degree polynomial over the finite field F32k , two conjectures on permutation trinomials over F32k proposed recently by Li, Qu, Li and Fu...

In this paper we are interested in to the divisibility of trinomials of type x am + x bs + 1 over a finite field GF (2). We show that, for any irreducible polynomial T of degree r over GF (2) and for any non-zero integers a and b, if there exist integers m and s such that T divides x am + x bs + 1 then a and b are not divisible by (2 r − 1).

This paper introduces a new approach for implementing GF(2 m ) multiplication using Polynomial Residue Number Systems (PRNS). Irreducible trinomials are selected as the generating polynomials for the PRNS channels to enable conversion to-and-from PRNS to be implemented using simple XOR networks. A novel approach for modular reduction over GF(2 m ) is also presented for the PRNS architecture to ...