Algebra Qualifying Examination
January 7, 2013
Instructions:
•There are 8 problems worth a total of 100 points. Individual point values are listed
next to each problem number.
•Credit awarded for your answers will be based on both the correctness of your answers
as well as the clarity and main steps of your reasoning. Answers must be written in a
structured and understandable manner and must be legible.
•You may use a calculator to check or perform simple computations, but it may not be
used as a step in your reasoning.
•Every effort is made to ensure that there are no typographical errors or omissions.
If you suspect there is an error, please check with the exam administrator. Do not
interpret a problem in a way that makes it trivial.
•Start each problem on a new page.
Notation: Throughout, let Zdenote the ring of integers; let Q,R, and Cdenote the fields
of rational, real, and complex numbers respectively; let Fqdenote the finite field with q
elements, where qis a power of a prime number. For a Galois extension of fields L/K, let
Gal(L/K) denote its Galois group.
1. [10 points] Let Gbe a group of order 56 = 23·7. Show that Gis not simple.
2. [10 points] Let Gbe a group of order 200 = 23·52, and let S8be the symmetric group
on {1,...,8}. Show that there exists a group homomorphism ψ:G→S8with proper
non-trivial kernel. (Hint: Find a set with 8 elements on which Gacts.)
3. [15 points] Give examples of the following objects. Be sure to verify that your examples
satisfy the desired properties.
(a) An irreducible polynomial over Qthat is irreducible by Eisenstein’s criterion for
p= 5.
(b) A unique factorization domain that is not a principal ideal domain.
(c) A finite extension of the rational function field Fp(x), for pa prime, that is normal
but not separable.
4. [10 points] Let Rbe a commutative ring with 1 6= 0. Let Mand Nbe left R-modules
such that Mis finitely generated and Nis noetherian. Show that M⊗RNis noetherian.
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