Department of Mathematics and Statistics Preliminary Ph.D. Examination in Algebra January 18, 1994

1. (a) Let G be a non-abelian group of order 8. Prove that there exists a cyclic subgroup

H of order 4 in G. Determine whether this subgroup H is normal in G.

(b) Give an example of a group of order 8 which does not admit any cyclic subgroup

of order 4.

2. Let K be a finite field of cardinality q.

(a) Show that K has characterstic p for some prime p.

(b) Show that the group of ring automorphisms Aut(K) is cyclic of order log_{p} q.

3. Let A_{4} denote the group of “even” permutations of a set of cardinality 4. (a) Determine the 2-Sylow subgroup(s) of A_{4}.

(b) Let T be a 3-Sylow subgroup of A_{4}. Show that T is not normal in A_{4}.

(c) Show that any group of order 12 which does not have a normal 3-Sylow subgroup

is isomorhic to A_{4}.

4. Let Q denote the field of rational numbers. Let K denote the field obtained by adjoining all the complex third roots of 2 to Q.

(a) Determine the degree [K : Q].

(b) Determine the Galois group of the extension K/Q.

(c) Determine all the subfields of K.

5. Let A be a ring and 0 → K → F →^{π}M → 0 , 0 → K′ → F′→^{π′}M → 0 be two short exact sequences of left A-modules, where F and F′ are free A-modules. Show that F ⊕ K′ ≅ F′ ⊕ K as left A-modules as follows: Let Y = {(f, f′) ∈ F ⊕ F′|π(f) = π′(f′)}. Let g : Y → F by g(f, f′) = f and let g′ : Y → F′ by g′(f, f′) = f′. Show that

(i) g and g′ are surjective. (ii) Y ≅ F ⊕ ker(g) ≅ F′ ⊕ ker(g′). (iii) ker(g) ≅ K′ and ker(g′) ≅ K. Hence F ⊕ K′ ≅ F′ ⊕ K.

6. Let R be a commutative ring.

(a) Show that P is a projective R-module iff Hom(P, −) when applied to a short exact sequence of R-modules is exact.

(b) Prove that every free module is projective.

(c) Give an example of a projective which is not free.

(d) Classify all projective modules over principal ideal domains.

7. (a) Show that every commutative ring with identity has a maximal ideal. (b) Give an example of a ring with a unique maximal nonzero ideal.

(c) Give an example of a ring with a finite (> 1) number of maximal ideals.

(d) Give an example of a ring with an infinite number of maximal ideals.

8. Let M_{n}(F) denote the ring of n-by-n matrices with entries in a field F.

Let E(i, j) denote the matrix with (i, j)-entry equal to 1 and all other entries equal to 0. Let A be a matrix in M_{n}(F).

(a) Describe the result of multiplication of A by E(i, j) on the left and on the right respectively.

(b) Show that M_{n}(F) has no non-trivial proper two-sided ideals.