Year: 1999
Type: article
Abstract: this article we consider powers and polynomials in the ring Zm , where m 2 IIN is arbitrary, and ask for #reduction formulas". For example, for addition, multiplication and exponentiation, wehave the following well known reduction formulas: a + b # mod#a; m#+mod#b;m# mod m #1# a # b # mod#a; m# # mod#b; m# mod m #2# a b # mod#a; m# b mod m #3# It is much more di#cult to #nd reduction formulas which allow to reduce the exponent. Of course in general a b 6# a mod#b;m# mod m: #4# In the second section we will investigate for whichnumbers m the reduction formula #4# holds. In the third and the two following sections we will consider generalizations of Fermat 's little theorem and Euler's Theorem which allow to replace #in Zm # certain powers a b by a polynomial f#a# of degree deg#f# which is strictly less than b. Such formulas can be useful for di#erent reasons: From an algorithmic point of view, it is cheaper to compute the polynomial f#a# modulo m than the full power a b modulo m. On the other hand one may wish for algebraic reasons to replace an arbitrary polynomial g#a#by a polynomial of #xed #lower# degree #depending only on m but not on g# which is, as a function in Zm , identical to g #see Section 6#. In the last section, we address the question of the minimal degree e#m# such that every polynomial in Zm can be written as a polynomial of degree q #e#m#. We give a complete answer to this question by determining minimal #normed# nullpolynomials modulo m. Throughout this paper, we use the customary shorthand notation a j b for a; b 2 Z with b a 2 Z.We write a # b mod m for numbers a; b 2 Z; m 2 IIN, if m j a , b and we adopt the notation #a; b# for the greatest common divisor of a and b. Furthermore we denote bymod#a; m# the uniquely determined n...
Source: Elemente der Mathematik
Language: English
Topic: Rings, Modules, and Algebras
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: bronze
OpenAlex ID: W2087691147