final - F is a generator in the (multiplicative) group of...

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Spring 2010 MATH 4320: Final Exam Instructor: Yuri Berest The exam is due 6 pm, Thursday, May 20 . Please turn in your exam in 439 Mallot Hall. Problem 1. ( 35 points ) Let A be a commutative ring with 1. a . An element a A is called nilpotent if a n = 0 for some n N . Prove that the set < = < ( A ) of all nilpotent elements is an ideal in A and the quotient ring A/ < has no nonzero nilpotent elements. Give an example of A with < ( A ) 6 = { 0 } . (The ideal < ( A ) is called the nilradical of A .) b . Let = ( A ) be the intersection of all maximal ideals in A . Show that = ( A ) is an ideal in A . Prove that a ∈ = ( A ) if and only if 1 - ab is invertible for all b A . Conclude that < ( A ) ⊆ = ( A ) , where < ( A ) is the nilradical of A . Problem 2. ( 40 points ) Let A [ x ] be the ring of polynomials in an indeterminate x , with coefficients in a commutative ring A . Let f = a 0 + a 1 x + a 2 x 2 + . . . + a n x n A [ x ] . Prove that a . f is a unit a 0 is a unit in A and a 1 , a 2 , . . . , a n are nilpotent; b . f is a zero divisor there exists a 6 = 0 in A such that af = 0 . Problem 3. ( 35 points ) a. Prove that x 2 + 1 is irreducible over Z 3 . Conclude that F = Z 3 [ x ] / ( x 2 + 1) is a field. List all the elements of F . Show that the class of the polynomial x + 1 in
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Unformatted text preview: F is a generator in the (multiplicative) group of units of F , but the class of x is not. b. Let Z [ 1 2 ] be the smallest subring of Q containing Z and 1 2 . Let (2 x-1) be the ideal of Z [ x ] generated by the polynomial 2 x-1 . Show that Z [ x ] / (2 x-1) = Z [ 1 2 ] . Is the ideal (2 x-1) maximal in Z [ x ]? If not, nd an ideal I Z [ x ] such that (2 x-1) I Z [ x ] . Bonus Problem. a. Let f ( x ) k [ x ] be an irreducible polynomial with coecients in a eld k . Suppose that f ( x ) has a root of multiplicity greater than one in some extension of k . Prove that char( k ) = p for some prime p and f ( x ) = g ( x p ) for some g ( x ) k [ x ] . b. Find all the values of a Z for which the polynomial f ( x ) = x 5-ax-1 is not irreducible in Z [ x ] . For each such a nd the corresponding factorization of f ( x ) into a product of irreducible polynomials in Z [ x ] ....
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