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\begin{center}
  \huge{Algebra Syllabus}\\
\large{Last modified: January  2008}
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The student should easily be able to give all relevant definitions,
provide standard examples, state major theorems, and provide the ideas
behind their proofs.  Specific topics which will be covered include:


\paragraph{Groups:}
Homomorphism theorems; group actions, Cayley's theorem, class
equation; elementary permutation group theory; the Jordan-H\"older
theorem; Sylow theorems; free, abelian, and simple groups; direct and
semi-direct products; structure of finitely generated abelian groups;
basic category theory, especially the notions of universal objects such
as free objects and (co)products, as well as contravariant and covariant
functors between categories.

\paragraph{Rings:}
Unique factorization domains, principal ideal domains, polynomial
rings, irreducibility criteria; matrix rings; Noetherian rings and the
Hilbert basis theorem; ideal theory; rings of quotients and
localization.

\paragraph{Fields:}
Prime fields; characteristic of a field; finite, algebraic,
transcendental, normal, separable and inseparable extensions; finite
fields; algebraic closure of a field; Galois theory; detailed examples
of the Galois correspondence.

\paragraph{Modules and Linear Algebra:}
Modules; free and projective modules, tensor products of modules and
algebras, structure theory for finitely generated modules over PIDs;
Finite dimensional vector spaces; linear dependence and independence;
linear transformations; dual space; determinants; characteristic
values and canonical forms.

\subsubsection*{REFERENCES}

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\begin{multicols}{2}
  

\reference{Artin}{Algebra}

\reference{Dennis \& Farb}{Noncommutative Algebra}

\reference{Dummit \& Foote}{Abstract Algebra}

\reference{Hoffman and Kunze}{Linear Algebra}

\reference{Jacobson}{Basic Algebra I, II}

\reference{Knapp}{Basic \& Advanced Algebra}

\reference{Lang}{Algebra}

\reference{Rotman}{The Theory of Groups}

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