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MATH 731 Abstract Algebra II – Final Exam
Tuesday, May 12, 2015 Check that that you have all six pages - note that the pages are double-sided
- (8 points) Suppose that V is an F -vector space spanned by the non-zero vectors S = {v 1 ,... , vn}. Prove that a subset of S is a basis for V.
- (10 points) (i) Show that if Q[α] = {p(α) : p(x) ∈ Q[x]} is a field then α ∈ C∗^ is algebraic over Q.
(ii) Prove that if α is algebraic over Q with minimal polynomial m(x) then Q[α] is a field with
Q[α] ' Q[x]/(m(x)).
- (10 points) Suppose that W is a subspace of the F -vector space V. Prove that
dimF (V /W ) = dimF (V ) − dimF (W )
- (12 points) How many similarity classes of 5 × 5 matrices over Q have characteristic polynomial f (x) = (x − 1)^3 (x − 2)^2? Give the Jordan canonical form for each class.
- (37 points) Let α = 4
2 and K the splitting field of x^4 − 2. a) K = Q( ).
b) [K : Q] =
c) A basis for K over Q is: { }
d) Describe the elements of G, the group of automorphisms of K. |G| = , G '.
e) Let G 1 = AutQ(i)K, the group automorphisms of K fixing Q(i). |G 1 | = , G 1 '. Find the proper subgroups of G 1 and the corresponding fields between Q(i) and K.
f) Let G 2 = AutQ(√2)K, the group automorphisms of K fixing Q(
2). |G 2 | = , G 2 '.
Find the proper subgroups of G 2 and the corresponding fields between Q(
- and K.
g) The fixed field of the automorphisms of Q(α) is. Is Q(α) Galois over Q?
- (7 points) Explain how to make a field with 49 elements.
- (14 points) Suppose that R is a commutative ring with unity, M an abelian group under +, together with an operation rm ∈ M for all r ∈ R, m ∈ M. a) What properties make M a unitary left R-module?
b) What additional properties make M a free R-module?
c) Is M 2 (R) =
{[a b c d
]
: a, b, c, d ∈ R
a free R-module? If so what is its rank?
d) Suppose that I is a non-principal ideal of R. Show that I is not a free R-module.
