Powers and polynomials in Z_m

In this article we consider powers and polynomials in the ring Z_m, where m in N is arbitrary, and ask for ''reduction formulas''. Further we consider generalizations of Fermat's little theorem and Euler's Theorem which allow to replace (in Z_m) certain powers a^b by a polynomial f(a) of degree deg(f) which is strictly less than b. Finally, we address the question of the minimal degree e(m) such that every polynomial in Z_m can be written as a polynomial of degree q < e(m). We give a complete answer to this question by determining minimal (normed) null-polynomials modulo m.

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