Algebra

                     Algebra                              Qualifying Exam, September 1999

Instructions: Do as many problems as you can. Please justify your answers and

show your work. All rings are associative with 1.

1. Prove or disprove:

(a) If H and K are subgroups of nite index in a group G then H \ K also has

nite index in G.

Let f : R ! S be a ring homomorphism.

(b) If I is a prime ideal of S then f􀀀1(I) is a prime ideal of R.

(c) If I is a maximal ideal of S then f􀀀1(I) is a maximal ideal of R.

2. Let K be a normal subgroup of a nite subgroup G. Assume n = [G : K] is

relatively prime to jKj. Show that K = fxn : x 2 Gg. What if K is not normal in

G?

3. Prove: (a) If G is a group of order 12 then G is not simple.

(b) If G is a group of order p2q, where p and q are distinct primes, then G is not

simple.

4. If I and J are ideals of a ring R and I + J = R, prove that R=IJ is isomorphic

to R=I R=J.

5. Show that an element a of a commutative ring A is nilpotent (i.e., ak = 0 for

some positive integer k) if and only if a is contained in every prime ideal of A.

6. If M is a nitely generated module over a Noetherian ring and f : M ! M is

an epimorphism, prove that f is an automorphism.

7. prove that any square matrix (over any eld) is similar to its transpose.

8. Prove: (a) If G is a nite non-cyclic abelian group then there is a positive integer

k < jGj such that gk = e (e is the identity of G), for every g 2 G.

(b) Every nite multiplicative subgroup of the the group of non-zero elements of a

eld is cyclic.

9. Let E be the splitting eld of x4 􀀀 4x2 + 2 over Q. Find all intermediate elds

K between E and Q.

10. Let !n = e2i=n.

(a) Describe the action of Gal(Q(!n)=Q) on Q(!n).

(b) Show that any sub eld of Q(!n) is Galois over Q.

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ALGEBRA

                                                                ALGEBRA

                                                          Qualifying Exam

                                                                May 1999

Instructions: Do as many problems as you can. Please justify your answers and

show your work. All rings are associative with 1.

1. Prove or disprove:

(a) If every subgroup of a group G is normal, then G is abelian.

(b) If D is PID then so is D[x].

(c) If E=K and K=F are _nite Galois extensions then E=F is also

Galois.

2. Let p be a prime number and G be a group with jGj = pn.

Show:

(a) The center of G is non-trivial.

(b) For each k _ nG has a normal subgroup H of order pk.

(3) Prove:

(a) If G is a group of order 12 then G is not simple.

(b) If G is a group of order p2q, where p and q are distinct

primes, then G is not simple.

4. If I and J are ideals of a ring R and I+J = R, prove that R=I\J is isomorphic

to R=ILR=J.

5. Show that an element a of a commutative ring A is nilpotent (i.e., ak = 0 for

some positive integer k) if and only if a is contained in every prime ideal of A.

6. If M is a _nitely generated module over a Noetherian ring and f : M ! M is

an epimorphism, prove that f is an automorphism.

7. Classify, up to similarity, all 3 _ 3 matrices T satisfying T3 = T over an

arbitrary _eld F.

8. Prove:

(a) If G is a _nite non-cyclic abelian group then there is a positive integer

k < jGj such that gk = e (e is the identity of G), for every g _G.

(b) Every _nite multiplicative subgroup of the group of non- zero elements

of a _eld is cyclic.

9. Let E be the splitting _eld of x4 􀀀 x2 + 2 over Q. Find all intermediate _elds

K between E and Q.

10. Let E be a _nite extension of Fq (the _nite _eld with q elements). Show

that E is Galois over Fq and Gal(E=Fq) is cyclic.