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\begin{document}
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\begin{center}

  \huge{The Department of Mathematics}\\[0.1\baselineskip]
  \Large{2022--23--B term}\\[0.2\baselineskip]

\end{center}

\begin{description}
  \item[Course Name]
    Noncommutative algebra

  \item[Course Number]
    \LRE{201‭.2‭.5121}

  \item[Course web page]\mbox{}\\
    \url{https://www.math.bgu.ac.il//en/teaching/spring2023/courses/noncommutative-algebra}

  \item[Lecturer]
    Prof. Eitan Sayag,
    \nolinkurl{<sayage@bgu.ac.il>},
    Office 107

\item[Office Hours]
  \url{https://www.math.bgu.ac.il/en/teaching/hours}
\end{description}

\section*{Abstract}




\section*{Requirements and grading\footnote{Information may change during the first two weeks of the term. Please consult the webpage for updates}}






\section*{Course topics}

\begin{enumerate}
\item{} Basic Algebraic Structures:   rings, modules, algebras, the center, idempotents, group rings
\item{} Division Rings:   the Hamiltonian quaternions, generalized quaternion algebras, division algebras over $\mathbb{F}_q$, $\mathbb{C}$, $\mathbb{R}$, $\mathbb{Q}$ (theorems of Frobenius and Wedderburn), cyclic algebras, the Brauer--Cartan--Hua theorem
\item{} Simplicity and semi-simplicity: simplicity of algebraic structures, semi-simple modules, semi-simple rings, Maschke's theorem
\item{} The Wedderburn--Artin Theory: homomorphisms and direct sums, Schur's lemma, the Wedderburn--Artin structure theorem, Artinian rings
\item{} Introduction to Group Representations: representations and characters, applications of the Wedderburn--Artin theory, orthogonality relations,  dimensions of irreducible representations, Burnside's theorem
\item{} Tensor Products:  tensor products of modules and algebras, scalar extensions, the Schur index, simplicity and center of tensor products, the Brauer group, the Skolem--Noether theorem, the double centralizer theorem, maximal fields in algebras, reduced norm and trace, crossed products
\end{enumerate}

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\end{document}

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