This preview shows page 1. Sign up to view the full content.
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 ] ....
View Full Document
- Spring '08