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\title[101 Midterm Exam]{Math 101: Algebra I \\ Midterm Exam}
\date{11 October 2016}

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\begin{center}
Name \ \underline{\hspace{3in}}

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\textbf{Problem} & \textbf{Score} \\
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1 \\
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2 \\
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3 \\
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4 \\
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Total \ \underline{\hspace{1in}}
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\begin{problab}{1}
Let $G$ be a group.  Indicate if the following statements are true or false.  If true, give a proof; if false, give an explicit counterexample.

\begin{enumalph}
\item If $H,H' \trianglelefteq G$ and $G/H \simeq G/H'$, then $H \simeq H'$.

\vspace{2.5in}

\item If $H,H' \trianglelefteq G$ and $H \simeq H'$, then $G/H \simeq G/H'$.

\vspace{2.5in}

\item If $K,K'$ are groups and $G \times K \simeq G \times K'$, then $K \simeq K'$.

\end{enumalph}
\end{problab}

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\begin{problab}{2}
Let $R$ be a Euclidean domain with norm $N$.
\begin{enumalph}
\item Let 
\[ m=\min(\{ N(a) : a \in R, a \neq 0\}). \]
Show that every nonzero $a \in R$ with $N(a)=m$ is a unit in $R$.

\vspace{2.5in}

\item Deduce that a nonzero element of norm zero in $R$ is a unit; show by an example that the converse of this statement is false.

\vspace{2.5in}

\item Let $F$ be a field and let $R=F[[x]]$.  Show that $R$ is Euclidean.  What does part (a) tell you about $R^\times$?  What are the irreducibles in $R$, up to associates?
\end{enumalph}
\end{problab}

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\begin{problab}{3}
Let $F$ be a field and let $V=\Mat_{2\times 3}(F)$ be the $F$-vector space of $2\times 3$-matrices.  
\begin{enumalph}
\item The group $\GL_2(F)$ acts on $V$ by left multiplication.  For $M,M' \in V$, the relation $M \sim M'$ if and only if $M'=AM$ for some $A \in \GL_2(F)$ defines an equivalence relation on $V$.  

What are the equivalence classes (i.e., the \emph{orbits} of the action)?

\vspace{2.5in}

\item Show that this action $\GL_2(F) \circlearrowright V$ induces an injective group homomorphism 
\[ \phi:\GL_2(F) \hookrightarrow \Aut_F(V).  \]

\vspace{2.5in}

\item Under the isomorphism $\Aut_F(V) \simeq \GL_6(F)$ given by the basis of matrix units, describe $\phi$ explicitly.
\end{enumalph}
\end{problab}


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\begin{problab}{4}
For the purposes of this exercise, we say that an isomorphism of $F$-vector spaces is \emph{natural} if it does not depend on a choice of basis.  

Let $F$ be a field and let $V,W$ be finite-dimensional vector spaces over $F$.  Show that there is a (well-defined) natural isomorphism of $F$-vector spaces
\[ \phi:V^* \otimes_F W \xrightarrow{\sim} \Hom_F(V,W). \]
\end{problab}

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