Unformatted text preview: 6.6. UNIQUE FACTORIZATION DOMAINS 307 Corollary 6.6.11. The irreducible elements of ROExŁ are of two types: irre- ducible elements of R , and primitive elements of ROExŁ that are irreducible in FOExŁ . A primitive polynomial is irreducible in ROExŁ if, and only if, it is irreducible in FOExŁ . Proof. Suppose that f.x/ 2 ROExŁ is primitive in ROExŁ and irreducible in FOExŁ . If f.x/ D a.x/b.x/ in ROExŁ , then one of a.x/ and b.x/ must be a unit in FOExŁ , so of degree . Suppose without loss of generality that a.x/ D a 2 R . Then a divides all coefficients of f.x/ , and, because f.x/ is primitive, a is a unit in R . This shows that f.x/ is irreducible in ROExŁ . Conversely, suppose that f.x/ is irreducible in ROExŁ and of degree 1 . Then f.x/ is necessarily primitive. Moreover, by Gauss’s lemma, f.x/ has no factorization f.x/ D a.x/b.x/ in FOExŁ with deg .a.x// 1 and deg .b.x// 1 , so f.x/ is irreducible in FOExŁ . n Proof of Theorem 6.6.7 . Let g.x/ be a nonzero, nonunit element of...
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- Fall '08
- Algebra, Mathematical terminology, OEX, RŒx