Algebra Prelim, January 2012

Do all problems

1. Recall that a proper subgroup of the group G is a subgroup H of G with G≠H. 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 ζ_n = e^2πi/n, a primitive nth root of 1.

   (a)

   Prove that Q(ζ_p) = Q(ζ_2p).

   (b)

   Prove that 1 + x² + x⁴ + ... + x^2p-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.