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\noindent{\Large{\textsc{Modern Algebra II\hfill Assignment 2.1  \\ Fall 2019 \hfill Due September 11}}}

\large

\vspace{1cm}

\begin{exercise}
Let $p, q \in \Z$ and set
\[
R = \left\{  \left( \begin{matrix} a & -bq \\ b & a-bp \end{matrix} \right) \bigg| a,b \in \Z \right\}.
\]
Prove that $R$ is a subring of $M_2(\Z)$ isomorphic to $\Z[\alpha]$, where $\alpha \in \C$ is a root of $x^2 + px + q$ (see Exercise 1.1.5).
\end{exercise}

\begin{exercise}
Let $\varphi : R \to R'$ be a homomorphism of rings.  Prove that if $\operatorname{char} R \ne 0$, then $\operatorname{char} R'$ divides $\operatorname{char} R$.
\end{exercise}

\begin{exercise}
Lang: Exercise III.3.24.
\end{exercise}


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