Algebra Preliminary Exam, Fall 1981

Instructions: do all eight problems
Notation: = integers, = rational numbers

1. Let G be a finite group.
   1. Let A and B be subgroups of G such that and AB = G. Prove that .
   2. Let H be a subgroup of G such that [G:H] = 2 and let . If m is the number of conjugates of h in G and n is the number of conjugates of h in H, prove that either n=m or n=m/2.
2. If the order of G is 105 and H is a subgroup of G of order 35, prove that .
3. Let P be a nonnormal p-Sylow subgroup of the finite group G. If is the normalizer of P in G, prove that is nonnormal in G.
4. Let F₁ and F₂ be finite fields of orders q₁ and q₂.
   1. Prove that q_i is a power of a prime, say for i=1,2.
   2. If , prove that p₁=p₂ and that is a divisor of .
5. 1. Prove that x⁴-2 is irreducible over .
   2. Let K be the splitting field of x⁴-2. Prove that the Galois group of K over , , is of order 8.
   3. Exhibit the correspondence (given by the Fundamental Theorem of Galois theory) between the subgroups of and the intermediate fields between and K.
6. 1. State Nakayama's lemma.
   2. Let R be a local commutative ring with maximal ideal M. Let X be a finitely generated R-module. Show that if X/MX can be generated by n elements, then so can X.
7. 1. Construct an example of finitely generated nonzero abelian groups A and B so that .
   2. If A and B are finitely generated abelian groups such that and , prove that and .
8. Let R be a commutative ring and let A be an ideal of R. Define the radical of A, denoted , by for some positive integer n}. You may assume that is an ideal of R and that .
   1. If P is a prime ideal of R such that , prove that and as a consequence show that .
   2. Prove that is the set of nilpotent elements of R/A. (An element r is nilpotent if rⁿ = 0 for some positive integer n.)

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