Euler's Theorem for Polynomials

摘要

The similarity of the arithmetic of the integers and the arithmetic ofpolynomials suggests that an analog of Euler's Totient theorem for integers also holds of polynomials over a finite field. This theorem is stated and proved, and then some properties of the totient function for polynomials are derived. The related notions of the order of one polynomials modulo another relatively prime polynomial, and of the exponent of a polynomial, are investigated. Finally, examples are given which show how to apply these ideas to the factorization of polynomials over finite fields. (KR)