Algebra Prelim

January 1997

Provide proofs for all statements.

If necessary, you may use the results from earlier parts of a problem, even though you may not have successfully proved them.

1. How many Sylow 2-subgroups does S_{3} × S_{4} have? You must justify your count.

2. Let G be a group and let a, b ∈ G have orders 30 and 21 respectively. If ab = ba determine possible orders for the element ab.

3. If k is a field, argue that every ideal of the ring ZZ × k[x] is principle where x is an indeterminate.

4. Construct an explicit field K such that Q ⊂ K ⊂ C and [K : Q ] = 12. Justify your answer.

5. Explain how we know that the real number:

α = ^{11} \sqrt{\frac{^{3} \sqrt{3} + ^{5} \sqrt{7}}{^{17} \sqrt{2} + 1}}

is algebraic over Q . Determine, with proof, a number m such that the minimum polynomial satisfied by α has degree less than m.

6. Let V be a finite dimensional vector space over a field k. Let f, g be linear transformations from V to itself such that g is an isomorphism. Argue that every eigenvalue λ of f is also an eigenvalue of g ∘ f ∘ g^{−1}.

7. For each positive integer n, let α_{n} denote a primitive n-th root of unity. Argue that Q (α_{n}, α_{m}) = Q (α_{l}) where l = LCM (m, n).

Compute [Q (α_{12}) : Q ].

8. Let k be a finite field with q elements and let n be coprime to q − 1. For each a ∈ k, argue that the equation x^{n} − a has a unique solution in k. Hint: Consider a map from k to k defined by α ⟶ α^{n}.

9. Let R be a commutative ring with identity and let M be the free R-module R^{n}. For i from 1 to n, let e_{i} = (0, ⋯, 1, ⋯, 0) be the member of M with 1 in the i − th slot.

For i from 1 to n let

f_{i} = ∑_{j=1}^{n} a_{ij}e_{j}

and let A = (a_{ij}) be the resulting n × n matrix. Prove that M = Rf_{1} + ⋯ Rf_{n} if and only if A is an invertible matrix over R.

10. Suppose that G is a group with 48 elements and that it contains a normal subgroup of order 3. Argue that it contains a normal subgroup of order 24.