Algebra Preliminary Exam, Spring 1984
Do all problems
- Find the Galois group of x6 - 1 over
- Show that a semisimple right Artinian
ring without zero divisors is a division
ring. (A ring has no zero divisors if ab = 0 implies a = 0 or b = 0.)
- Show that there are no simple groups of order 300.
- Let R be a commutative domain with field of fractions F.
Prove that F is an injective R-module.
- State and prove a structure theorem analogous to the
Fundamental Theorem for modules over a PID, which describes finitely
(You may assume the Fundamental Theorem
for any argument.)
- Assume that K/F is a Galois field extension and
lies in an algebraic closure of K.
Prove that |Gal(K/F)| divides
deg a, where
denotes the degree of
- Prove that a finite p-group with a unique subgroup of index
p is cyclic. (Hint: first consider abelian p-groups.)
- Let f : M - > N be a surjective homomorphism of left
R-modules. Show that if P is a projective R-module, then the
induced map f* : Hom
R(P, M) - > HomR(P, N)
is a surjection of abelian groups.
- Let k be a field and let
R = k[X1,..., Xm] be the
polynomial ring in m indeterminates. Prove that if M is a simple
R-module, then dim
kM < .