test2 - Math 5125 Wednesday November 2 Second Test Answer...

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Math 5125 Wednesday, November 2 Second Test. Answer All Problems. Please Give Explanations For Your Answers 1. Suppose R = 0 is a commutative ring with a 1. Prove that the set of prime ideals in R has a minimal element with respect to inclusion. (17 points) 2. Let R be a UFD. Prove that R [ x ] has infinitely many irreducible elements, no two of which are associates. (17 points) 3. Prove that 16 x 4 + 16 x 3 - 4 x + 5 is irreducible in Q [ x ] (consider the automorphism induced by x ( x - 1 ) / 2 and then the prime 2). (16 points)
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Math 5125 Wednesday, November 2 Solutions to Second Test 1. Let A denote the set of prime ideals of R . Since R = 0, it has maximal ideals. Fur- thermore every maximal ideal is a prime ideal, consequently A = /0. Partially order the prime ideals of A by reverse inclusion; that is P Q means Q P . Suppose { P j | j J } is a chain in A (where J is an indexing set). Let Q = j P j . Then Q is cer- tainly an ideal of R (the intersection of ideals is always an ideal), so we need to check that it is prime. Suppose a , b R - Q . Then a / P
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