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 xⁱy^j 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 $\in$ R, the ideal pR + qR is principal. Prove that for every a, b $\in$ 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 $\otimes_{R}^{}$ 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 $\oplus$ M @ N $\oplus$ N. Prove that M @ N.

6. Let K be a field of characteristic zero, let f $\in$ K[x] be an irreducible polynomial, let L be a splitting field for f over K, and let a₁,a₂,a₃,a₄ $\in$ L be the roots of f. Suppose [L : K] = 24 (i.e. dim_KL = 24).

   (i)

   If 1 < i, j < 4 are integers, prove that L =/= K(a_i,a_j).

   (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,..., x_n]). Suppose U /\ V = f. Prove that I(U) + I(V) = k[x₁,..., x_n] (where I(U) is the set of all polynomials in k[x₁,..., x_n] which vanish on U).