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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 x1) be the ideal of Z [ x ] generated by the polynomial 2 x1 . Show that Z [ x ] / (2 x1) = Z [ 1 2 ] . Is the ideal (2 x1) maximal in Z [ x ]? If not, nd an ideal I Z [ x ] such that (2 x1) 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 5ax1 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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 Spring '08
 CRISSINGER
 Math

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