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\begin{document}
\textbf{Math 6310, Assignment 6 \hfill Due in class: Monday, November 30}

% \medskip {\large Name: }

\begin{enumerate}

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\item Determine, up to isomorphism, all finitely generated subgroups of $\QQ$.

%\textbf{Solution:}


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\item Determine all maximal ideals of the ring $\ZZ[x]/(120,x^3+1)$.

%\textbf{Solution:}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item Let $M$ be a $\ZZ$-module that is a homomorphic image of $\ZZ^2$, such that each element of $M$ is annihilated by $6$. Determine all such $M$ up to isomorphism.

%\textbf{Solution:}

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\item Let $\FF$ be a field, and $R=\FF[x]$ a polynomial ring. Let $M$ be the quotient of $R^3$ by the submodule generated by the elements $(x^2,x^3,x^4)$ and $(1,x+1,x^2)$ of $R^3$. Express $M$ as a direct sum of cyclic $R$-modules.

%\textbf{Solution:}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item If $M$ is a square matrix over $\CC$ with $M^3=M$, prove that $M$ is diagonalizable. Is this true over arbitrary fields?

%\textbf{Solution:}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item Find the Jordan form of the matrix
\[
\begin{pmatrix}
1&1&1\\
1&1&2\\
0&0&0\\
\end{pmatrix}.
\]

%\textbf{Solution:}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item Find the possible Jordan forms of a real $3\times 3$ matrix $M$ satisfying $M^2+M+I=0$.

%\textbf{Solution:}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item Let $a,b,c$ be real numbers with $a^2+b^2+c^2=1$. Determine the possible Jordan forms of the matrix
\[
\begin{pmatrix}
0 & a & b \\ -a & 0 & c \\ -b & -c & 0 \\
\end{pmatrix}.
\]

%\textbf{Solution:}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item Determine the number of conjugacy classes of elements of order $2$ in $\GL_n(\FF)$. Does this depend on the characteristic of the field $\FF$?

%\textbf{Solution:}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item If $M$ is a $5\times 5$ complex matrix with $\det(M)=1$, $\trace(M)=5$, $\rank(M-I)=3$, and minimal polynomial having degree $3$, find the possible Jordan forms of $M$.

%\textbf{Solution:}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{enumerate}
\end{document}

