6.8. IRREDUCIBILITY CRITERIA 313 is an ideal in R , and that A k ± A k C 1 for all k ²0 . 6.7.2. Suppose that R is Noetherian and ' W R ! S is a surjective ring homomorphism. Show that S is Noetherian. 6.7.3. We know that Z Œ p ³ 5Ł is not a UFD and, therefore, not a PID. Show that Z Œ p ³ 5Ł is Noetherian. Hint: Find a Noetherian ring R and a surjective homomorphism ' W R ! Z Œ p ³ 5Ł . 6.7.4. Show that Z ŒxŁ C x Q ŒxŁ is not Noetherian. 6.7.5. Show that a Noetherian domain in which every irreducible element is prime is a unique factorization domain. 6.8. Irreducibility Criteria In this section, we will consider some elementary techniques for determin-ing whether a polynomial is irreducible. We restrict ourselves to the problem of determining whether a polyno-mial in Z ŒxŁ is irreducible. Recall that an integer polynomial factors over the integers if, and only if, it factors over the rational numbers, according to Lemma 6.6.9 and Corollary 6.6.10 . A basic technique in testing for irreducibility is to reduce the polyno-
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Prime number, Integral domain, Ring theory, Polynomial ring, Noetherian