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Algebra Qualifying Exam September 20, 2014 Do all five problems. 1. Let G be a finite abelian group of odd order. Let φ : G → G be the function defined by φ(g) = g 2 for all g ∈ G. Prove that φ is an automorphism. 2. Let G be a group and suppose that H ≤ G. The normalizer of H in G is defined to be N (H) = {g ∈ G|gH = Hg} and the centralizer of H in G is defined to be C(H) = {g ∈ G|gh = hg for all h ∈ H}. (a) Prove that N (H) is a subgroup of G. (b) Prove that C(H) is a normal subgroup of N (H) and that N (H)/C(H) is isomorphic to a subgroup of Aut(H). 3. Let V denote the real vector space of polynomials in x of degree at most 3. Let B = {1, x, x2 , x3 } be a basis for V and T : V → V be the function defined by T (f (x)) = f (x) + f 0 (x). (a) Prove that T is a linear transformation. (b) Find [T ]B , the matrix representation for T in terms of the basis B. (c) Is T diagonalizable? If yes, find a matrix A so that A [T ]B A−1 is diagonal, otherwise explain why T is not diagonalizable. 4. Let f (x) = x3 + x + 1 ∈ Z5 [x]. (a) Prove that f (x) is irreducible. (b) Prove that hf (x)i is a maximal ideal. (c) What is the cardinality of Z5 [x]/hf (x)i? Justify. 5. Let R be a commutative ring R. The nilradical of R is defined to be N = {r ∈ R|rn = 0 for some n ∈ N}. (a) Prove that N is an ideal of R. (b) Prove that N is contained in the intersection of all prime ideals of R.