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 \title{\vspace{-.5in} Midterm \hspace{1in} TMath 403 \hspace{1in}  Spring 2026}
\date{}

\begin{document}
\maketitle
 \vspace{-0.9in}

\section{Directions:} 

\noindent Use any resources available to prepare, including your classmates, the TLC, internet, or me.  You should plan, discuss, and debate answers with anyone that is willing to engage.  {\bf{You are required to cite your sources and collaborators.}}

\begin{packed_itemize}
\item Sign up for a 25 minute interview slot at: \url{https://calendly.com/r-e-vanderpool/403-midterm}
\item The assessment is closed book but you make use a one-sided 8.5" by 11" sheet of notes with definitions and theorems.  No prepared solutions.
\item Location: SNO 236B

\item Please arrive 5 minutes early and get your materials prepared so that we can start promptly.  You will have a maximum of 25 minutes for the midterm.


%\item Remote
%\begin{packed_itemize}
%\item For your interview, please sign into the zoom room at least 5 minutes before your selected time.  Have your working surface clear of clutter with only the allowed materials, blank paper, computer/tablet/phone.
%\item You need a working camera to give a tour of your space.  I would appreciate being able to see your face during the interview, but that is not required. 
%\item You will have a maximum of 25 minutes to include a brief tour of your space and discussion/solutions to three questions
%\item You are required to share your screen when asked.  Failure to comply will result in 0.
%\end{packed_itemize}

\end{packed_itemize}

\hrulefill


\subsection{Grading Rubric}


\noindent The exam consists of three questions. The following rubric will be used for each question.


\begin{tabular}{ |c|l| } 
 \hline
 95\% &Well-executed. Thorough discussion. All points are well supported. \\ 
 	&Two or fewer minor errors.  No nontrivial errors.\\
 \hline
 85\% & Generally well-executed. Several minor errors; or a nontrivial mathematical \\ 
 	& error that gets corrected when identified.\\
 \hline
75\% & Uncorrected nontrivial error; or several nontrivial errors that get \\
	&corrected when identified; or error in fundamental understanding that\\
	&gets corrected when identified.\\
 \hline
60\% & Error in understanding of fundamental concept that does not get corrected.\\
 \hline
0 & No evidence of preparation or understanding.\\	 
 \hline
\end{tabular}

\begin{packed_itemize}
\item (20 pts) The first question will be {\bf{your choice}}.
\item (20 pts) I choose the second question from the ones remaining.
\begin{packed_itemize}
\item You may pass on my choice once for a 5 percentage point penalty on the overall Midterm score.  If the pass is used, I select another problem.
\end{packed_itemize}
\item (10 pts) Last problem that has an element of chance in it.

\end{packed_itemize}

\newpage

\section{Midterm Questions:} 


\begin{enumerate}
\item  Determine if the following algebraic objects satisfy algebraic conditions enumerated below. Be prepared to explain your choices with examples (SLO \#4). If none of the objects satisfy the given algebraic condition, be prepared to provide a different example, if possible.


$\mathbb{Z}_{14} \times \mathbb{Z}_7$ \hspace{1in} $\mathbb{Q}[[x]]$ \hspace{1in} $\mathbb{Z}_3 \times \mathbb{Z}_3$ \hspace{1in} $\mathbb{Z}_{29}$ \hspace{1in} 

\begin{enumerate}
\item Does not have a commutative ring structure.
\item Has a subgroup that is not an ideal.
\item Is an Integral Domain.
\item Is an infinite ring with at least 3 zero divisors.
\item Is not a field.
\end{enumerate}


%involves DHW1Tue & DHW1Thur
\item Prove or provide a counterexample for the following statement: ``The ideal $I=(x, 5)$ is maximal in $\mathbb{Z}[x]$."


%DHW3Thur irreducibility and DHW3Tue max ideal so factor is field. DHW4Tue prime for ID
\item Let $f(x)\in \mathbb{F}_3[x]$. We will examine $\mathbb{F}_3[x] / (f(x))$ performing addition \& multiplication between elements as well as determining what kind of algebraic object it is (field? ID?).

To define $f(x)$, use the first three letters of your first name to build $f(x)$ of the form $a_0 + a_1x + a_2x^2$. Specifically, use the table below to let $a_0$ be the number
that corresponds to the first letter in your first name. For Ruth then "R" would set
$a_0 = 0$. Let $a_1$ be the number that corresponds to the second letter and $a_2$ correspond
to the third letter. For Ruth then $a_1 = 1$ and $a_2 = 2$, thus the polynomial for Ruth is
$f(x)=0+ 1x +2x^2$.

\begin{tabular*}{3.2in}{l|lllllllllllll}
	1 &A & D &G&J&M&P&S&V&Y &L & U\\

	2 &B & E &H&K&N&Q&T&W&Z &C &I &O \\

	0 & & F && & &R& &X& \\
	\end{tabular*}	

%DHW4Thur
\item Let $R=\mathbb{Z}[\sqrt{3}]$ and Frac$(R)$ be the field of fractions of $R$. Describe the elements in Frac$(R)$ and be prepared to add, subtract, multiply, and divide different elements from Frac$(R)$. 
\end{enumerate}

%DHW2Tue
\noindent Everyone will do the last question which examines maps from $\mathbb{Z}_{10 \cdot a}$ to $\mathbb{Z}_{10\cdot b}$. You will role a die to determine $a$ and I will roll the die to determine $b$. The questions will be largely impromptu investigating SLO \#2, but below are some of the questions to give you an idea of what might be asked:

\begin{enumerate}
\item Build a map and verify it is a well-defined ring homomorphism.
\item Can we build a one-to-one ring homomorphism between the rings that is not onto?
\item Can we build an onto ring homomorphism between the rings that is not one-to-one?
\item For a ring homomorphism $\phi$, what is the factor ring ($\mathbb{Z}_{10\cdot a} / \ker (\phi)$) isomorphic to?
\end{enumerate}



\end{document}
