Paper 2015/926

CRITERION OF MAXIMAL PERIOD OF A TRINOMIAL OVER NONTRIVIAL GALOIS RING OF ODD CHARACTERISTIC

Vadim N. Tsypyschev and Julia S. Vinogradova

Abstract

In earlier eighties of XX century A.A.Nechaev has obtained the criterion of full period of a Galois polynomial over primary residue ring modulo power of 2. Also he has obtained necessary conditions of maximal period of the Galois polynomial over such ring in terms of coefficients of this polynomial. Further A.S.Kuzmin has obtained analogous results for the case of Galois polynomial over primary residue ring of odd characteristic . Later the first author of this article has carried the criterion of full period of the Galois polynomial over primary residue ring of odd characteristic obtained by A.S.Kuzmin to the case of Galois polynomial over nontrivial Galois ring of odd characteristic. Using this criterion as a basis we have obtained criterion calling attention to. This result is an example how to apply results of the previous work of V.N.Tsypyschev in order to construct polynomials of maximal period over nontrivial Galois ring of odd characteristic. During this it is assumed that period of polynomial modulo prime ideal is known and maximal .

Note: Theorem 2.2 provides by method how to verify in easy way whenever a polynomial of special form over nontrivial Galois ring R has a maximal period. Let's note here once more that we don't concern the task of evaluating period of its image modulo pR. We suggest that its period modulo pR is maximal as a predefined condition. And after that we concern period of investigating polynomial over ring R. From other side the same Theorem provides an easy way to construct polynomials of maximal period of special form over nontrivial Galois ring $R$.