Ph.D. Preliminary Examination in Algebra

June 4, 1999

1. Let A be an n × n matrix with entries in the field C of complex numbers that satisfies the relation A^{2} = A. Show that A is similar to a diagonal matrix which has only 0’s and 1’s along the diagonal.

2. Furnish examples of the following:

(a) A finite group that is solvable but not abelian.

(b) A finite group whose center is a proper subgroup of order 2.

(c) A nested sequence of finite groups G, H, K with H a normal subgroup of G and K a normal

subgroup of H such that K is not a normal subgroup of G.

3. Let p be the polynomial p(t) = t^{5} +t^{2} +1 regarded as an element of the ring A = F_{2}[t] of polynomials with coefficients in the field F_{2} of two elements. Show that p is irreducible, and then find a polynomial of degree at most 4 with the property that its residue class modulo the ideal pA generates the entire multiplicative group of units in the quotient ring A/pA.

4. Let G be a finite group of order N, and let n be a positive integer that divides N. Do one of the following:

(a) Prove that if G is abelian, then G contains a subgroup of order n.

(b) Find an example of G, N, n as above where G has no subgroup of order n.

5. Show that every group of order 30 contains a normal cyclic subgroup of order 15.

6. Let F be the field Q(i) where i = \sqrt{−1} ∈ C , and let E be the splitting field over F of the polynomial f(t) = t^{4} − 5 . Find:

(a) the extension degree [E : F].

(b) the group Aut_{F} (E) of all automorphisms of E that fix F.

7. Let F_{2} be the field of 2 elements, and let R be the commutative ring

R = F_{2}[t]/t^{3}F_{2}[t] .

(a) How many elements does R contain?

(b) What is the characteristic of R?

(c) Find all ring homomorphisms R → R.

8. Let a, b, c, d be elements of a field F, let A, B, C, D be n × n matrices over F, and let

m = ( \begin{matrix}a & b \\ c & d \end{matrix} ) and M = ( \begin{matrix}A & B \\ C & D \end{matrix} ) .

If λ : F^{2} → F^{2} and Λ : F^{2n} → F^{2n} denote the linear endomorphisms corresponding (relative to standard coordinates) to m and M, respectively, then to what linear endomorphism that may be constructed from λ and Λ may one relate the 4n × 4n (Kronecker product) matrix

( \begin{matrix}aA & bA & aB & bB \\ cA & dA & cB & dB \\ aC & bC & aD & bD \\ cC & dC & cD & dD \end{matrix} ) ?

Explain your answer.