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Unformatted text preview: MTHSC 851/852 (Abstract Algebra) Dr. Matthew Macauley HW 14 Due Friday, October 9, 2009 (1) Prove Proposition 2.6 from lecture: (a) If F E K and E is stable, then G E G. (b) If H G, then F H is stable. (2) For each field extension, compute the degree, give a basis, and find the Galois group. 4 (a) Q( 2) over Q (b) Q( 2, 3, i) over Q (c) Q( 3 2, ) over Q, where is a primitive third root of unity. (d) Q() over Q, where is a primitive nth root of unity. (e) A degree-n extension of a finite field Fq (where q = pk ), over Fp . (3) Let = 3 + 3 2 R and K = Q(). (a) Find [K : Q]. (b) Let f (x) be the minimal polynomial for over Q, and G be the Galois group of f (x) over Q. Find the order of G. (4) Suppose that F K is a field extension of degree n < and E is any field containing F . (a) Prove that there are at most n distinct F -homomorphisms : K E (i.e., (x) = x for all x F ). (b) Show that if E is algebraically closed, there exists at least one F -homomorphism K E. (c) Show that if n = p is prime, then there need only exist one F -homomorphism K E. (5) Let K/F be a Galois extension of degree 2n , and suppose that char(F ) = 2. Show that there exists a chain of intermediate subfields F = M0 M1 Mn-1 Mn = K such that Mi = F (ai ), where a2 Mi-1 . i (6) Prove that (Q, +) is not isomorphic to the Galois group of any algebraic field extension ...
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