# Math 817: Problem set 5

Instructions: Write up any four of the following problems (but you should try all of them, and in the end you should be sure you know how to do all of them). Your write ups are due Friday, October 1, 2004. Each problem is worth 10 points, 9 points for correctness and 1 point for communication. (Your goal is not only to give correct answers but to communicate your ideas well. Make sure you use good English, so proofread your solutions. Once you finish a solution, you should restructure awkward sentences, and strike out anything that is not needed in your approach to the problem.)
1. Let e denote the base of the natural logs (as usual). You may assume the facts that e is transcendental (i.e., there is no nontrivial polynomial f(x) with rational coefficients for which f(e) = 0) and that the square root of 2 is not a rational number. Let s denote the square root of 2. Regarding the reals R as a vector space over the rationals Q, then V = Span({1, e}) and W = Span({1,s}) are Q-subspaces. With respect to the usual operation of multiplication in R, prove that W is in fact a field, but that V is not.
2. Prove that every solution of 2x1 - x2 - 2x3 = 0 has the form (1.5) on p. 80. I.e., every solution is a linear combination of w1 and w2, where the components of the former are 1, 0 and 1, and those of the latter are 1, 2 and 0.
3. Define a concept of homomorphism of fields in such a way that if f: F -> K is a homomorphism of fields, then f is injective and Im(f) is a field. Your definition may not explicitly include the requirement that f be injective or that Im(f) be a field. Then prove that f is injective and Im(f) is a field using your definition.
4. Let A be the 2x2 matrix 8e1,1 + 3e1,2 + 2e2,1 + 6e2,2, let x be the 2x1 column vector x1e1,1 + x2e2,1 and let b be the 2x1 column vector 3e1,1 - e2,1. (Compare #10, p. 104.)
• (a) Solve Ax = b in Fp, for p = 2, 5, and 7.
• (b) Determine the number of solutions when p = 7.
5. Let p be a positive integer which is a prime. (See #14, p. 105.)
• (a) Let n be any integer. If p does not divide n, prove that np-1 = 1 (mod p).
• (b) Let n be any integer. Prove that np = n (mod p).
6. Let p be a positive integer which is a prime. (See #15, p. 105.)
• (a) Prove that the product of all nonzero elements of Fp is -1.
• (b) Prove Wilson's Theorem that (p - 1)! = -1 (mod p).