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Algebra Qualifying Examination January 7, 2008 Instructions: • There are eight Algebra questions worth a total of 100 points. Individual point values are listed next to each problem. • Credit awarded for your answers will be based on the correctness of your answer as well as the clarity and main steps of your reasoning. “Rough working” will not be accepted: answers must be written in a structured and understandable manner. • You may use a calculator to check your computations (but you will not earn points for using it as a step in your reasoning). Notation: Throughout, Z denotes the integers, Q denotes the rational numbers, R denotes the real numbers, and C denotes the complex numbers. 1. (15 points) Let n be a positive integer. Denote by Sn and An the symmetric and alternating groups of degree n, respectively. (a) For each positive integer n, give a composition series of Sn . (b) Find the centralizer in S5 of the 3-cycle (345). (c) Find the centralizer in A5 of the 3-cycle (345). 2. (15 points) List all possible groups of order 175 (up to isomorphism). 3. (10 points) Let R be a ring, S a subring of R, and I an ideal of R. Suppose that R = S + I and S ∩ I = 0. (a) Prove that each element of R may be written r = s + i for unique elements s ∈ S and i ∈ I. (b) Prove that R/I is isomorphic to S by giving a ring isomorphism from R/I to S. 4. (10 points) Let R1 and R2 be integral domains with fields of fractions (i.e. quotient rings) F1 and F2 , respectively. Suppose that there is an injective ring homomorphism φ : R1 → R2 . Prove that φ extends to F1 , i.e. there is an injective ring homomorphism ψ : F1 → F2 such that ψ(r) = φ(r) for all r ∈ R. 5. (10 points) Let R be a ring. Let P1 and P2 be R-modules. Prove that P1 ⊕ P2 is a projective R-module if, and only if, both P1 and P2 are projective. 6. (10 points) Let A be a finitely generated abelian group. (a) If A is finite, prove that A ⊗Z Q = 0. (b) If A is infinite, prove that A ⊗Z Q ∼ = Qr , an isomorphism of Z-modules, for some positive integer r. 7. (15 points) √ √ (a) Q Express the minimal polynomial f (x) of 2 + 3 over Q as a product (x − αi ), for complex numbers αi . √ √ (b) Q Express the minimal polynomial g(x) of 3 2 + 3 4 over Q as a product (x − βi ), for complex numbers βi . (c) What is the Galois group of f (x)g(x) over Q? √ √ 8. (15 points) Let W = Q( 2, −2). √ √ (a) Extend { 2, −2} to a basis of W as a vector √ space over Q. (b) Let f : W → W be defined by f (α) = (1 + 2)α. Verify that f is a linear transformation of W as a vector space over Q and find its characteristic polynomial. (c) Find a basis of eigenvectors for f of C ⊗Q W .