Algebra Prelim, Fall 2003

Do all problems

1. Prove that a group of order 2256 = 47*48 cannot be simple.

2. Let G = < x, y | x7 = y3 = 1, yxy-1 = x2 >.
(i)
Prove that every element of G can be written in the form xiyj where i, j are non-negative integers.

(ii)
Prove that G has order at most 21.

(iii)
Prove that there is a homomorphism q : G - > S7 such that

qx = (1 2 3 4 5 6 7), qy = (2 3 5)(4 7 6).

(iv)
Prove that G has order 21.

3. Let R be a UFD. Suppose that for every coprime p, q R, the ideal pR + qR is principal. Prove that for every a, b R, the ideal aR + bR is principal. (Coprime means that the greatest common divisor of p, q is 1.)

4. Let R be the ring Z/4Z and let M be the ideal 2Z/4Z. Prove that M M @ M as R-modules.

5. Let R be a PID and let M, N be R-modules. Suppose M is finitely generated and M M @ N N. Prove that M @ N.

6. Let K be a field of characteristic zero, let f K[x] be an irreducible polynomial, let L be a splitting field for f over K, and let a1,a2,a3,a4 L be the roots of f. Suppose [L : K] = 24 (i.e. dimKL = 24).
(i)
If < i, j < 4 are integers, prove that L =/= K(ai,aj).

(ii)
Prove that L = K[a1 + 2a2 + 3a3].

7. Let k be an algebraically closed field, let n be a positive integer, and let U, V be affine algebraic sets in kn (so U is the zero set of a collection of polynomials in k[x1,..., xn]). Suppose U /\ V = f. Prove that I(U) + I(V) = k[x1,..., xn] (where I(U) is the set of all polynomials in k[x1,..., xn] which vanish on U).

Peter Linnell
2003-08-04