Unformatted text preview: 11. An integral domain R is a UFD if and only if every nonzero nonunit element is a product of irreducibles, and irreducibles are prime. 12. A PID is a UFD, but not conversely. 13. R [ x ] is a UFD when R is a UFD 14. Let R be an integral domain and let p be a prime in R . Then p is a prime in R [ x ] . 15. Suppose R is a UFD with quotient ﬁeld K , f is a primitive polynomial in R [ x ] and g is an arbitrary polynomial in R [ x ] . If f ± ± g in K [ x ] , then f ± ± g in R [ x ] . 16. Gauss’ lemma 17. Let R be a UFD with ﬁeld of fractions K , and let f ∈ R [ x ] be primitive with degree at least 1. Then f is irreducible in R if and only if f is irreducible R [ x ] . 18. If F is a commutative ring and f ∈ F [ x ] , then f ( a ) = 0 if and only if xa is a factor of f . 19. Eisenstein’s criterion....
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 Fall '07
 PALinnell
 Math, Algebra, Prime number, Integral domain, UFD

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