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Graduate Qualifying Exam in Algebra School of Mathematics, University of Minnesota Fall 2006 You may use any well known results that do not trivialize the problem in the opinion of the examiners. If you use such a result, you must explain exactly how you are applying it. Unjustified or inadequately justified answers will receive no credit. In questions with several parts, credit will be obtained if only some of the parts are completed. Be sure to put your code number (not your name) on each bluebook that you hand in. The questions are divided into two Sections, A and B. Answer all the questions, from both sections, but write answers to questions from Section A in a different set of blue books to the questions from Section B, labelling each blue book ‘Section A’ or ‘Section B’. SECTION A In this section each question is worth 10 points. 1. Let G be a cyclic group of order 12. Show that the equation x5 = g is solvable for every g ∈ G, and that the solution x is unique (for a given g ∈ G). 2. Let ζ be a primitive complex 9th root of unity. Find all intermediate fields between Q and Q(ζ), and give explicit expressions for generators for them. 3. Let k be a field with 16 elements. Show that the polynomial x4 + x3 + x2 + x + 1 factors into linear factors in k[x]. 4. In the ring C[x, y], find all prime ideals containing both x3 and y 3 . 5. Let R be a commutative ring whose elements are linear endomorphisms of a finitedimensional complex vector space (that is, linear maps from the space to itself). Show that there is a common eigenvector for all the operators in R. PLEASE TURN OVER 1 SECTION B 6. (13 points) Let G be the group of matrices G= a b 0 c a, b, c ∈ Z/3Z, a 6= 0 6= c . Determine whether G is isomorphic to any of the groups A4 , D12 or C2 × C6 , that is, the alternating group of degree 4, the dihedral group of order 12, and the direct product of a cycle of order 2 and a cycle of order 6. 7. (13 points) (a) (4 points) Let A and B be commutative rings with identity and consider the ring A ⊕ B where the addition and multiplication are defined componentwise in the usual way: (a1 , b1 ) + (a2 , b2 ) := (a1 + a2 , b1 + b2 ) (a1 , b1 )(a2 , b2 ) := (a1 a2 , b1 b2 ). Show that every ideal in A ⊕ B has the form I ⊕ J where I is an ideal of A and J is an ideal of B. (b) (9 points) Now let K be a field and let R be the factor ring R = K[X]/(X 2(X +1)). Determine how many ideals of R are projective as R-modules, giving generators for each such ideal. 8. (12 points) Let α be a complex number which is algebraic over Q and let L be a subfield of C which is a splitting field for α over Q. Let g be any non-identity element of G = Gal(L/Q). (a) (6 points) Show that there is an element h ∈ G so that (hgh−1 )α 6= α. (b) (6 points) Show that L is generated as a field by {gα g ∈ G}. [You may assume the fundamental theorem of Galois theory.] a b 9. (12 points) Let H = be a matrix with a, c ∈ R and b ∈ C, and let b c f u1 u2 v1 , = ( u1 v2 u2 ) H v1 v2 be the corresponding Hermitian bilinear form on C2 . Prove that f is positive definite if and only if a > 0 and ac − |b|2 > 0. 2