QUALIFYING EXAM IN ALGEBRA August 2011 1. There are 18 problems on the exam. Work and turn in 10 problems, in the following categories. I. Linear Algebra 1 problem II. Group Theory 3 problems III. Ring Theory 3 problems IV. Field Theory 3 problems 2. Turn in only 10 problems. No credit will be given for extra problems. All problems are weighted equally. 3. Put each problem on a separate sheet of paper, and write only on one side. Put your name on each page. 4. If you feel there is a misprint or error in the statement of a problem, then interpret it in such a way that the problem is not trivial. 1
I. Linear Algebra 1. Let A be an n n Jordan block. Prove that if B is an n n matrix that commutes with A, then B is a polynomial in A. 2. Let A = 2 0 0 0 1 1 1 1 0 0 1 0 1 0 0 2. (a) Find the characteristic polynomial of A. (b) Find the minimal polynomial of A. (c) Find the dimensions of all eigenspaces of A. (d) Find the Jordan canonical form of A. 3. Let A, B be complex 3 3 matrices having the same eigenvectors. The minimal polynomial of A is (x 1) 2 and the characteristic polynomial of B is x 3. Show that the minimal polynomial of B is x 2. 2
II. Group Theory 1. Let G be a group with the property that each conjugacy class in G has size at most 2. Prove that G Z(G). 2. Show that if G is a non-abelian simple subgroup of the symmetric group S n, then G is contained in the alternating subgroup A n. ( HINT: Consider the index G : G A n ) 3. Prove or disprove: every group of order 14077 = 7 2011 is cyclic. Use Sylow s theorems. 4. Let g and h be elements of the alternating group A n which have the same cycle structure. Assume that in a cycle decomposition of g (and hence of h), two cycles have the same length. Prove that g and h are conjugate in A n. 5. Let G be a finite group which has a maximal, simple subgroup H. Prove that G is either simple, or else there exists a minimal normal subgroup N of G such that G/N is simple. 3
III. Ring Theory 1. Let R be any ring with identity, and n any positive integer. If M n (R) denotes the ring of n n matrices with entries in R, prove that M n (I) is an ideal of M n (R) whenever I is an ideal of R, and that every ideal of M n (R) has this form. 2. Let R be a commutative ring with 1 such that for every x in R there is an integer n > 1 (depending on x) such that x n = x. Show that every prime ideal of R is maximal. 3. Let R be a commutative ring with 1 and let f(x) R[x] be nilpotent. Show that the coefficients of f are nilpotent. ( HINT: Show first that if f, g are nilpotent, then so is f ± g. ) 4. Let R S be commutative domains with the same identity, and assume that S is an integral extension of R. Let I be a nonzero ideal of S. Prove that I R is a nonzero ideal of R. 5. Let F be a field. Prove that the polynomial ring F[x] is a PID, and that F[x, y] is not a PID. 4
IV. Field Theory 1. Determine the Galois group of x 4 4 over Q and identify all the intermediate fields between Q and the splitting field of x 4 4. Use the Fundamental Theorem. 2. (a) Prove that if E is a finite dimensional extension of F which is generated over F by a set S of elements a satisfying a 2 F for all a S, then E : F = a power of 2. (b) Give an example that shows 2 cannot be replaced by 3. 3. Let E = F(α) be a simple extension of F and fix β E F. Prove that α is algebraic over F(β). 4. Let K be a Galois extension of the field F with Galois group G. Let g(x) be a monic polynomial in F[x] and suppose g(x) splits over K (that is, K contains a splitting field for g(x) over F) and let K be the set of roots of g(x). Prove that g(x) is a power of an irreducible polynomial over F if and only if G is transitive on. 5. Let E = Q[α, β] where α 2 Q, β 2 Q, and E : Q = 4. If γ E Q and γ 2 Q prove that γ is a rational multiple of one of α, β, or αβ. (The Fundamental Theorem may be useful.) 5