Algebra Qualifying Examination
12 January 2010
Instructions:
•There are eight problems worth a total of 100 points. Individual point values are
listed by each problem.
•Credit awarded for your answers will be based upon the correctness of your answers
as well as the clarity and main steps of your reasoning. “Rough working” will
not receive credit: Answers must be written in a structured and understandable
manner.
•You may use a calculator to check your computations (but may not be used as a
step in your reasoning).
•All rings have an identity and all modules are unital.
Notation: Throughout Qdenotes the rationals and Aut(K/Q) denotes the group of field
automorphisms of Kfixing Q.
1. (21 points) The exponent exp(G) of a group Gis the smallest k∈ {1,2,...}∪{∞}
such that gk=efor all g∈G.
(a) Show that a finitely generated abelian group Awith exp(A)<∞is finite.
(b) Give an example of an infinite group of finite exponent.
(c) Give an example of a group Gin which every element has finite order but
exp(G) = ∞.
2. (18 points) Let T∈End(V) be a linear operator on a vector space Vwith
dim(V) = nsuch that minT(x) = charT(x), i.e. the minimal and characteris-
tic polynomials of Tcoincide.
(a) Show that there exists an α∈Vsuch that {α, T (α),...,Tn−1(α)}is a basis
for V.
(b) Show that if U∈End(V) satisfies UT =T U then Uis a polynomial in T.
3. (8 points) Let Rbe a principal ideal domain. Show that if P6= (0) is a prime
ideal then Pis maximal.
4. (10 points) An R-module Mis indecomposable if there are no R-submodules A6= 0
and B6= 0 of Msuch that M=A⊕B. Show that if Ris a principal ideal domain
then if Mis indecomposable either M∼
=Ror M∼
=R/(pn) for some prime element
pof R.
5. (7 points) Show that a group of order 80 cannot be simple.
6. (21 points) Find the degree of the splitting field (over Q) of the following polyno-
mials:
(a) x3−x−2
(b) (x2−2)(x2−5)
(c) (x2−2)(x2−5)(x2−10)
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