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 degreen 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 Mn1 Mn = K such that Mi = F (ai ), where a2 Mi1 . i (6) Prove that (Q, +) is not isomorphic to the Galois group of any algebraic field extension ...
View
Full Document
 Fall '08
 Staff
 Algebra, Galois theory, Galois group, Matthew Macauley HW

Click to edit the document details