Algebra Qualifying Examination
11 August 2011
Instructions:
•There are 8 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).
•Every effort is made to ensure that there are no typographical errors or omissions.
If you suspect there is an error, check with the exam administrator. Do not
interpret the problem in a way that makes it trivial.
Notation: Throughout, Qand Cdenote the field of rational or complex numbers, re-
spectively. Zdenotes the ring of integers and Fdenotes the field with two elements.
1. (12 points) Show that any group of order 455 is cyclic.
2. (13 points) Decompose 35 into product of prime elements of Z[i] (and show this
is indeed a prime decomposition).
3. (11 points) Suppose that [Q(u) : Q] is odd. Show that Q(u2) = Q(u).
4. (11 points) Find an inverse of (1 + x)3in F2[[x]].
5. (13 points) Let Mbe a module over a ring R, N and Psubmodules in M. Define
(N:P) = {r∈R|rP < N}. Show that (N:P) is an ideal of R. Show also that
(N:P) = Ann((N+P)/N), where if Lis an Rmodule then Ann(L) = {r∈
R|rL = 0}.
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