{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

College Algebra Exam Review 303

College Algebra Exam Review 303 - 6.8 IRREDUCIBILITY...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
6.8. IRREDUCIBILITY CRITERIA 313 is an ideal in R , and that A k ± A k C 1 for all k ² 0 . 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 is not a UFD and, therefore, not a PID. Show that Z Œ p ³ is Noetherian. Hint: Find a Noetherian ring R and a surjective homomorphism ' W R ! Z Œ p ³ . 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-
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

Ask a homework question - tutors are online