Algebra Prelim

January 1996

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. Prove that all groups of order 2n − 1 are abelian for 1 ≤ n ≤ 10.

2. Let G be the subroup of the permutation group S_{6} generated by a and b, where

a = (123456), b = (16)(25)(34).

• Show that A =< a > is a normal subgroup of G.

• Find the order of G.

• Let Z denote the center of G. Show that Z ⊆ A. Find Z.

• Let G′ denote the commutator of G. Show that G′ ⊆ A. Find G′.

• Find a subgroup H of G such that G ≅ H × Z.

3. Let M_{n}(k) denote the vector space of n × n matrices over a field k. Show the following: • Determine dim_{k}(M_{n}(k)).

• The set of all upper triangular matrices forms a subspace of M_{n}(k). Determine its dimension.

• Consider the set {I, A, A^{2}, ⋯, A^{r}, ⋯}, where I stands for the identity matrix. Prove that at most n^{2} members of S are linearly independent.

• Do you know a number m < n^{2} such that at most m members of the set are linearly independent. Give a proof or cite appropriate theorem(s). (Note that we are asking for a bound which is valid for all matrices A of size n × n.)

4. Let S = k[X]/(g(X)) be the quotient ring of the polynomial ring k[X] over a field k by the principal ideal (g(X)). Assume that the polynomial g(X) is of the form X^{n} + a_{1}X^{n−1} + ⋯ + a_{n} with n > 0.

• Prove that S is naturally a vector space over k and calculate its dimension.

• Prove that S is an integral domain iff g(X) is irreducible in k[X].

• Prove that every element of S is algebraic over k. Guess and prove an upper bound for the algebraic degree of all possible elements of S.

5. Give an example with proof of an integral domain D and an element x ∈ D such that x is irreducible but not prime. (A prime element is one which generates a prime ideal.)

6. Let k be a field. Let J = {f(X, Y) ∈ k[X, Y]|f(0, 0) = f(2, 3) = 0}. Determine all the maximal ideals containing J. Is J a prime ideal? Why? Is J a maximal ideal? Why?

7. Let F be a finite field of characteristic p. Prove the following statements.

• The map ϕ : F ⟶ F defined by ϕ(x) = x^{p} is an automorphism.

• Every x ∈ F has a unique p-th root in F.

• Let X be an indeterminate over F and let \overline{F} denote an algebraic closure of F. If f(X) ∈ F [X] is an irreducible polynomial of degree n > 0. Then there exists some α ∈ \overline{F} such that

f(X) = ∏_{i=0}^{n−1}(X − α^{p^{i}})

• F is a perfect field, i.e. every irreducible polynomial in one variable over F is separable. (Define a separable polynomial while you are at it!)

8. Prove that the polynomial Y^{3} − X^{5} − 1 is irreducible over Q [X, Y]. How, if at all, does your conclusion change if the field Q is changed to some other field?