Algebra Prelim

January 2000

Provide proofs for all statements.

If necessary, you may use the results from other parts of this test, even though you may not have successfully proved them.

In the following, Q denotes the field of rational numbers and ZZ denotes the ring of integers.

All 10 questions carry equal weight in determining your grade.

1. Let G be a group of order pq where p, q are primes such that p < q and such that GCD(p, q−1) = 1. Prove that G is cyclic.

2. Let G be a group of order p^{2} where p is prime. Prove that G is abelian.

3. Argue that there is no simple group of order 132.

4. Let S be the subgroup of the abelian group ZZ × ZZ generated by (6, 9) and (5, 5).

Briefly explain why the quotient group ZZ × ZZ/S is a direct sum of finitely many cyclic groups.

Further, find the cyclic groups which form the direct sum in this case.

5. Let R be a commutative ring with identity 1 ≠ 0. Prove that R is a field if and only if {0} and R are the only ideals of R.

6. Let R be an integral domain and let g(X) ∈ R[X]. Find necessary and sufficient conditions on the polynomial g(X) such that the homomorphism ϕ defined by ϕ(f(X)) = f(g(X)) is an automorphism of R[X].

Give an example of a polynomial g(X) for which the corresponding ϕ is an automorphism of R[X] but is different from the identity homomorphism. Justify your claim.

Give an example of a polynomial g(X) for which the corresponding homomorphism ϕ fails to be an automorphism. Justify your claim.

7. Let f : V ⟶ W be a linear transformation of vector spaces of equal finite dimensions.

Prove the Theorem that the following are equivalent:

(a) f is an isomorphism.

(b) The kernel of f is {0}.

(c) f is surjective.

Show, by an example, that the theorem fails if dimV >dimW. You must justify your example and explain exactly what fails.

8. Show that f(X) = X^{5} + 9X + 12 is irreducible in Q [X] (the polynomial ring with rational coefficients). Let θ be a root of f(X). Show that the field Q (θ) is a finite dimensional vector space over Q . calculate its dimension. Find the inverse of 1 + θ in the field Q (θ).

9. Let f(X) = X^{n} + a_{n−1}X^{n−1} + ⋯ + a_{1}X + a_{0} ∈ ZZ[X]. If α ∈ Q is a root of f(X), prove that α ∈ ZZ and further that α divides a_{0} in ZZ.

10. Let f(X) = X^{4} − 2 ∈ Q [X]. Then (a) Determine a splitting field K for f(X) over Q .

(b) Find the degree [K : Q ] of the extension field K over Q .

(c) Compute, with justification, the galois group of K over Q .