Computational Commutative Algebra by Kreuzer and Robbiano

By Kreuzer and Robbiano

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However, in Tutorial 6 we give some hints on how to do it for univariate polynomials over Z/(p) . Other notions which everyone learns in school are the least common multiple and greatest common divisor of natural numbers. 8). Other subjects related to the unique factorization property will be considered in the exercises and tutorials. Now, let us do first things first and introduce the notions of irreducible and prime elements in such a way that it is possible to speak about factorizations. 1. Let R be an integral domain and r ∈ R \ {0} be a nonunit.

By possibly clearing the denominators, we see that there exists an 34 1. Foundations element r ∈ R such that rf = g2 h2 with g2 , h2 ∈ R[x]. 10 we know that r = cont(g2 ) · cont(h2 ) . Thus we can simplify and get a new equation f = g3 h3 with primitive polynomials g3 , h3 ∈ R[x]. Since the degrees of g3 and h3 are positive and R is an integral domain, neither is a unit, contradicting the irreducibility of f . Thus we have shown that f is irreducible in Q(R)[x]. 2). Consequently, the polynomial f is prime in Q(R)[x].

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