Algebra Prelim, August 2011

Do all problems

1. Prove that there is no simple group of order 380.

2. Let k be a field with | k| = 7, let n be a positive integer, and let f, g be coprime polynomials in k[x1,..., xn]. If f3 - g3 = h3 for some nonzero polynomial hk[x1,..., xn], prove that there exists pk[x1,..., xn] and uk such that f - g = up3. Hint: factor f3 - g3 as a product of three polynomials, and note that these polynomials are pairwise coprime.

3. Let R be a PID, let p be a prime in R, and let M, N be finitely generated left R-modules such that pMpN. Assume that if 0≠mM or N and pm = 0, then Rm is not a direct summand of M or N respectively (i.e. there is no submodule X such that RmX = M or N). Prove that MN.

4. Let f (x)∈Q[x] be an irreducible polynomial of degree 9, let K be a splitting field for f over Q, and let α∈K be a root of f. Suppose that [K : Q] = 27. Prove that Q(α) contains a field of degree 3 over Q.

5. Let p be a prime and let A denote all pn-th roots of unity in C. Thus A is the abelian subgroup of the nonzero complex numbers under multiplication defined by {e2πim/pn | m, nN}, in particular A is a Z-module. Determine AZA.

6. Let k be a field, let A∈M3(k), the 3 by 3 matrices with entries in k, and suppose the characteristic polynomial of A is x3. Prove that A has a square root, that is a matrix B∈M3(k) such that B2 = A, if and only if the minimal polynomial of A is x or x2.

7. Let R be an integral domain, let n be a positive integer, let S be a subset of the polynomial ring in n variables R[x1,..., xn], and define Z(S) = {(r1,..., rn)∈Rn | f (r1,..., rn) = 0 for all fS}, the zero set of S. Prove that there exists a finite subset T of S such that Z(S) = Z(T).

Peter Linnell 2011-08-13