Algebra Qualifying Exam
September 18, 2015
Do all five problems.
1. Determine the number of group homomorphisms φ, between the given
groups. Here, K4 denotes the Klein four-group (also known as Z2 × Z2 )
and S3 denotes the symmetric group on 3 elements.
(a) φ : K4 → Z2
(b) φ : Z2 → K4
(c) φ : S3 → K4
(d) φ : K4 → S3
2. (a) Show that if G is a group (not necessarily finite) and H is a subgroup
of G, then G is a disjoint union of the left cosets of H.
(b) State and prove Lagrange’s Theorem for (finite) groups.
3. Let R be an integral domain. Suppose that a and b are nonassociate irre­
ducible elements in R, and the ideal (a, b), generated by a and b, is a proper
ideal. Show that R is not a principal ideal domain (PID).
4. Let F be a field and let α be an element that generates a field extension of
F of degree 5. Prove that α2 generates the same extension.
⎡
⎤
−2
1 −1
2 ⎦.
5. Let A = ⎣ 5 −2
7 −3
3
(a) Find the characteristic polynomial and the minimal polynomial of A.
(b) Find the Jordan canonical form of A.