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# rev21 - 11 An integral domain R is a UFD if and only if...

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Math 5125 Monday, October 31 Second Test Review Test on Wednesday, November 2. One of the problems will be identical to one of the un- graded homework problems (since ﬁrst test, but not on presentations, September 30), and one of the problems will be identical to one of the sample test problems. The test will cover Chapters 7,8, and Sections 9.1–4. Topics will include 1. Fundamental homomorphism theorem and ideal correspondence theorem. 2. In a commutative ring R , the ideal P is prime if and only if R / P is an integral domain, and P is maximal if and only if R / P is a ﬁeld. 3. Zorn’s lemma and maximal ideals. 4. Rings of fractions and universal property. 5. Polynomial rings and universal property. 6. Division algorithm. 7. F [ x ] is a PID when F is a ﬁeld. 8. In an integral domain, primes are always irreducible. 9. If 0 6 = p R and R is an integral domain, then p is prime if and only if ( p ) is a prime ideal. 10. In a PID, a nonzero prime ideal is maximal.
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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 x-a is a factor of f . 19. Eisenstein’s criterion....
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