advertisement

Algebra Qualifying Examination January, 2006 Directions: 1. Answer all questions. (Total possible is 100 points.) 2. Start each question on a new sheet of paper. 3. Write only on one side of each sheet of paper. Policy on Misprints: The Qualifying Exam Committee tries to proofread the exams as carefully as possible. Nevertheless, the exam may contain a few misprints. If you are convinced a problem has been stated incorrectly, indicate your interpretation in writing your answer. In such cases, do not interpret the problem in such a way that it becomes trivial. Notes: 1. All rings are unitary. All modules are unitary. 2. Q is the rationals, R the reals, C the complexes, and Z the integers. 1. (10 points) Let R be a ring with 1. R may be regarded as a left R-module over itself, R R. Prove that a submodule M of R R is a direct summand of R R if and only if there is an element e ∈ R such that e2 = e and M = Re. 2. (15 points) Let D be the subring of Z[x] generated by 1, x2 , x3 , i.e. D = h{1, x2 , x3 }i. a. Describe the elements of D. b. Show that x2 ∈ D is irreducible, but not prime. c. What ring is D/(x2 )? 3. (9 points) Prove that the additive group (Q , +) of rational numbers is not a direct sum of two proper subgroups. 4. (8 points) Let m and n be positive integers. Let R be the ring Zmn. Let I = (m), the ideal of R generated by m ∈ R. Prove that R/I is isomorphic to Zm as rings. 5. (8 points) Let G be a finite group. Let H be a normal subgroup of G. Suppose p is a prime integer that does not divide [G : H]. Prove that H contains every Sylow p-subgroup of G. 6. (10 points) Let F be a field. Recall that an algebraic extension K of F is a normal extension of F if every irreducible polynomial in F [x] that has one root in K actually splits in K[x]. Let K be a normal extension of the field F of finite degree. Let E be a subfield of K containing F . Prove that E is a normal extension of F if and only if every F -isomorphism of E into K is an F -automorphism of E. 1 7. (15 points) Let V = P2 (R ), the vector space of polynomials over the reals R of degree 2 or less. Define T : V → V by T (f (x)) = f (0) + f (1)(x + x2 ). T is a linear transformation on V . a. Find the characteristic polynomial for T . b. Find a basis for V consisting of eigenvectors for T . c. Find T 8 (1 + x). 8. (10 points) Let p be an odd prime integer. Let F = GF (p) be the finite field of p elements and K = GF (p3 ), the finite field with p3 elements. a. Explain why every irreducible polynomial of degree 3 in F [x] factors completely in K[x]. b. Explain why no irreducible polynomial of degree 2 in F [x] factors in K[x]. 9. (15 points) Let f (x) = x4 − 4 ∈ Q [x]. Let K be the splitting field for f (x). Let G be the Galois group of f (x). a. Describe the elements in K. b. List the elements in G. c. Identify the group G. 2