Algebra Prelim, January 2012

Do all problems

1. Recall that a proper subgroup of the group G is a subgroup H of G with GH. Now suppose G is a finite cyclic group. Prove that G is not a union of proper subgroups.

2. Prove that a group of order 6435 = 9·5·11·13 cannot be simple.

3. Prove that M2(Q)⊗M2(Z)M2(Q)≌M2(Q) as (M2(Q), M2(Q))-bimodules ( M2(Q) indicates the ring of 2 by 2 matrices with entries in Q).

4. Let R be a PID which is not a field and let M be a finitely generated injective R-module. Prove that M = 0.

5. Let p be an odd prime and for a positive integer n, let ζn = e2πi/n, a primitive nth root of 1.
(a)
Prove that Qp) = Q2p).

(b)
Prove that 1 + x2 + x4 + ... + x2p-2 is the product of two irreducible polynomials in Q[x].

6. Determine the isomorphism class of the Galois group of the polynomial x5 - 5x - 1 over Q.

7. For n a positive integer, let An denote affine n-space over Q.
(a)
Prove that every element of Q[x, y]/(x3 -y2) can be written in the form (x3 -y2) + f (x) + yg(x) where f (x), g(x)∈Q[x].

(b)
Prove that Q[x, y]/(x3 -y2)≌Q[t2, t3], the subring of the polynomial ring Q[t] generated by t2, t3.

(c)
Prove that Q[t2, t3] is not a UFD.

(d)
Let V denote the affine algebraic set Z(x3 -y2), the zero set of x3 -y2 in A2. Determine the coordinate ring of V.

(e)
Is V isomorphic to A1 as affine algebraic sets? Justify your answer.

Peter Linnell 2012-01-11