Algebra Preliminary Exam, Spring 1984

Do all problems

1.

Find the Galois group of x⁶ - 1 over $\mathbb {Q}$ .

2.

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.)

3.

Show that there are no simple groups of order 300.

4.

Let R be a commutative domain with field of fractions F. Prove that F is an injective R-module.

5.

State and prove a structure theorem analogous to the Fundamental Theorem for modules over a PID, which describes finitely generated $\mathbb {Z}$/n$\mathbb {Z}$ -modules. (You may assume the Fundamental Theorem for any argument.)

6.

Assume that K/F is a Galois field extension and a lies in an algebraic closure of K. Prove that |Gal(K/F)| divides |Gal(K(a)/F(a))| deg a, where deg a denotes the degree of a over F.

7.

Prove that a finite p-group with a unique subgroup of index p is cyclic. (Hint: first consider abelian p-groups.)

8.

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) - > Hom_R(P, N) is a surjection of abelian groups.

9.

Let k be a field and let R = k[X₁,..., X_m] be the polynomial ring in m indeterminates. Prove that if M is a simple R-module, then dim _kM < $\infty$ .