Algebra Qualifying Exam
March 14, 2015
Do all five problems.
1. Let G and H be groups of order 10 and 15, respectively. Prove that if there
is a nontrivial homomorphism φ : G → H, then G is abelian.
2. Let G be an abelian group and GT be the set of elements of finite order in
G.
(a) Prove that GT is a subgroup of G.
(b) Prove that every non-identity element of G/GT has infinite order.
(c) Characterize the elements of GT when G = R/Z where R is the additive group of real numbers.
3. (a) Suppose I and J are ideals in a commutative ring with unity R such
that R = I + J. Prove that the map f : R 7→ R/I × R/J given by
f (x) = (x + I, x + J) induces the isomorphism
R/IJ ∼
= R/I × R/J.
(b) Prove that Z3 [x] /(x3 − x2 − 1) ∼
= Z3 [x] /(x3 + x + 1). (Hint: Use part
(a).)
4. An element r of a ring R is said to be idempotent if r2 = r. Suppose that
R is a commutative ring with unity containing an idempotent element e.
(a) Prove that 1 − e is also idempotent.
(b) Prove that eR and (1 − e)R are both ideals in R and that
R∼
= eR × (1 − e)R.
(c) Prove that if R has a unique maximal ideal, then the only idempotent
elements in R are 0 and 1.
5. Suppose T is a linear transformation on a finite dimensional complex inner
product space V . Let I denote the identity transformation on V . The
numerical range of T is the subset of C defined by
W (T ) := {hT (x), xi : x ∈ V, kxk = 1}.
(a)
(b)
(c)
(d)
Show that W (T + cI) = W (T ) + c for every c ∈ C.
Show that W (cT ) = cW (T ) for every c ∈ C.
Show that the eigenvalues of T are contained in W (T ).
Let B be an orthonormal basis for V . Show that the diagonal entries
of [T ]B are contained in W (T ).