Algebra Prelim

June 2002

Provide proofs for all statements, citing any theorems that may be needed. 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, ZZ denotes the ring of integers and C the field of complex numbers.

1. Let G be a 3-group acting on a set S of 24 elements. Suppose that there is an element s_{1} ∈ S such that g · s_{1} = s_{1} ∀g ∈ G.

Prove that there is s_{2} ∈ S distinct from s_{1} such that g ·s_{2} = s_{2} ∀g ∈ G.

2. Prove that the polynomial f(X) = X^{3} + 6X + 2 is irreducible over Q . Let θ be a root of f(X) in C . Determine (with proof) the minimum polynomial of −θ over Q . Determine (also with proof) the minimum polynomial for 1 − θ.

3. Let G be a group and let H_{1}, H_{2} be two distinct maximal subgroups of G. If K is a normal subgroup of H_{1} as well as H_{2}, then argue that K is a normal subgroup of G.

Discuss if the above conclusion can hold if H_{1} = H_{2}.

4. Let I be the ideal (2 − X, X + 3) in ZZ[X].

(a) Prove that I is a prime ideal.

(b) Prove that I is not a principal ideal.

(c) Determine I ∩ ZZ.

5. Let A, B be n × n matrices over a field k with n ≥ 1 and let their columns be denoted as A_{1}, ⋯, A_{n} and B_{1}, ⋯, B_{n} respectively. Assume that they have the same null space, i.e.

AX = 0 iff BX = 0 ∀X ∈ k^{n}.

(a) Construct an example of a pair of such matrices over a field of

your choice, satisfying:

• A and B have dimension at least two.

• A and B are not scalar multiples of each other.

• A, B satisfy the stated condition.

(b) If 1 ≤ r ≤ n prove that A_{1}, ⋯, A_{r} are linearly independent iff B_{1}, ⋯, B_{r} are linearly independent.

(c) Prove that A, B have the same rank.

6. Let F_{p} be a finite field with p elements and let f ∈ F_{p}[X] be an irreducible polynomial of degree n. Argue that if α is a root of f in some extension field L, then α, α^{p}, ⋯, α^{p^{n−1}} are all the distinct roots of f in L. Using this or otherwise, argue that F_{p}(α) is a Galois extension of F_{p} and determine its Galois group.

7. Let p, q be distinct primes in Q . Let L = Q (\sqrt{p}, \sqrt{q}).

(a) Prove that [L : Q ] = 4.

(b) Determine the Galois group of L over Q .

(c) Explicitly find an element θ ∈ L such that L = Q (θ).

8. Let g(X) be an irreducible monic polynomial of degree n > 1 over Q .

(a) Prove that there is a polynomial h_{1}(Y) ∈ Q [Y] of degree n such that h_{1}(2X) is divisible by g(X). Explain if possible how to find it concretely using the given g(X).

(b) Prove that there is a polynomial h_{2}(Y) ∈ Q [Y] of degree n such that h_{1}(X^{2}) is divisible by g(X). Explain if possible how to find it concretely using the given g(X).

(c) As an illustration, or practice, do the above exercise for the concrete polynomial X^{3} − 2.