\vsize 9in
\hsize 6.5in
% \magnification \magstep 1
\def\Hom{\operatorname{Hom}}
\def\Q{{\Bbb Q}}
\def\Z{{\Bbb Z}}
\def\goodbreak{\par\penalty-500 }
\def\problem#1\par{\bigskip\bigskip\goodbreak\centerline{Problem #1}\bigskip}
\font\bigbf=cmbx12
\def\ni{\noindent}
\def\bs{\bigskip}
\def\ms{\medskip}
\def\ss{\smallskip}
\def\subhead{\bf}
\centerline{ \bigbf Spring Semester, 1990 }
\bs
\centerline { \bigbf Math 402 -- Second Course in Abstract Algebra, II}
\ms
\centerline { \bigbf Prof. Daniel Grayson }
\ms
\centerline { 2:00 PM MWF, Section F1, 243 Altgeld Hall }
\bs
\bs

The course will be organized as a mixture of lectures and problem solving in
class, with the emphasis on the latter.  Students will be responsible for
reading substantial amounts of material.

\bs

{\bf Text}: {\sl Algebra} by Hungerford.

\bs

\centerline{Syllabus}


\parindent = 0pt
\def\in{\par\advance \parindent by 24pt \hangindent \parindent \hangafter 1}
\def\out{\par\advance \parindent by -24pt}

\list
\item"I." Basic concepts
\inlevel
\item"A." Partially ordered sets, totally ordered sets, Zorn's lemma
\item"B." Well-ordered sets, ordinals, cardinals
\item"C." Category theory, product, coproduct, additive category, direct sum,
		adjoint functors, universal properties and uniqueness.
\outlevel
\item"II." Linear algebra and diagonalization of matrices.
\inlevel
\item"A." Modules, presentations of modules, homomorphisms, matrices, tensor product.
\item"B." Duality for finite abelian groups with $\Hom(G,\Q/\Z)$.
\item"C." Homomorphisms of free modules over a principal ideal domain, Smith
		normal form, Fitting ideals.
\item"D." Endomorphisms of vector spaces, rational canonical form, Jordan normal form.
\item"E." Bilinear forms on vector spaces.
\item"F." Inner products, law of inertia.
\item"G." Hermitian matrices (diagonalization, eigenvalues are real).
\outlevel
\item"III." Multilinear algebra.
\inlevel
\item"A." Symmetric powers, symmetric algebra.
\item"B." Exterior powers, exterior algebra, determinants.
\outlevel
\item"IV." Polynomials in several variables.
\inlevel
\item"A." Hilbert basis theorem, noetherian rings.
\item"B." Hilbert Nullstellensatz (the easy proof over uncountable fields).
\item"C." Groebner bases; algorithm for deciding whether a system of
		polynomial equations has a solution.
\item"D." Transcendence degree (aiming toward dimension of algebraic varieties).
\outlevel
\item"V." Module theory.
\inlevel
\item"A." Projective modules, injective modules, exact functors, left-exact
		functors, right-exact functors.
\item"B." Filtrations, Schreier-Zassenhaus theorem.
\item"C." Composition series, Jordan-Holder theorem for modules and the analogue for groups.
\item"D." Krull-Schmidt theorem for modules.
\outlevel
\item"VI." Noncommutative algebra.
\inlevel
\item"A." Semisimple rings and modules, Wedderburn theory.
\item"B." Maschke's theorem, representations (and characters) of finite groups.
\outlevel
\endlist
\end


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Here we set the local variables for my emacs text editor.
%   Local Variables:
%   TeX-command: "lamstex"
%   TeX-start-of-header: "%**start of header"
%   TeX-end-of-header:   "%**end of header"
%   End:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
