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 A2 = 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) = t5+t2+1 regarded as an element of the ring A = F2[t] of
polynomials with coefficients in the field F2 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
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 =
−1 ∈ C , and let E be the splitting field over F of
the polynomial f(t) = t4 − 5 . Find:
(a) the extension degree [E : F ].
(b) the group AutF (E) of all automorphisms of E that fix F .
7. Let F2 be the field of 2 elements, and let R be the commutative ring
R = F2[t]/t3F2[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
and M =
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 endo-
morphism that may be constructed from λ a