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 \title{\vspace{-.5in} Final \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 two-sided 8.5" by 11" sheet of notes with definitions and theorems.  No prepared solutions.
\item Location: 
\begin{packed_itemize}
\item Thursday June 4th: SNO 222C
\item Friday June 5th: MDS 303C
\item Tuesday June 9th: SNO 236B
\end{packed_itemize}

\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 final.


\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 Final exam 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{Final 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.


$\mathbb{Z}_{14}$ \hspace{1in} $\mathbb{Q}(\sqrt[3]{11})$ \hspace{1in} $\mathbb{Z}_5 \times \mathbb{Z}_{15}$ \hspace{1in} $\mathbb{Z}_{29}[x]$ \hspace{1in} 

\begin{enumerate}
\item Is a commutative ring but not a principal ideal domain.
\item Is a ring without unity.
\item Is a Euclidean domain but not a field.
\item Is an finite ring with at least 4 zero divisors.
\item Is a field.
\end{enumerate}


%involves Formal WHW7
\item Let $n$ be the number of letters in your first name and $m$ be the number of letters in your last name. For example, "Ruth" would correspond to $n=4$. 
\begin{enumerate}
\item Describe $\mathbb{Q}(\sqrt{n}+\sqrt{m})$ (SLO \#3).
\item Find $[\mathbb{Q}(\sqrt{n}+\sqrt{m}):\mathbb{Q}]$.
\end{enumerate}



\item Define the set of constructible numbers, $C$, any binary operators, and identify what kind of algebraic object it is. Be prepared to provide examples of numbers that are in $C$ and examples of numbers that are not.


\item Let $F=\mathbb{Q}(\sqrt{2},\sqrt{5})$. We write an element of $F$ with the basis $\{1, \sqrt{2}, \sqrt{5}, \sqrt{10}\}$. \\Define $\tau:F\rightarrow F$ by $\tau(a+b\sqrt{2}+c\sqrt{5}+d\sqrt{20})=a-b\sqrt{2}+c\sqrt{5}-d\sqrt{10}$ for $a,b,c,d\in \mathbb{Q}$. Define $\sigma:F\rightarrow F$ by $\sigma(a+b\sqrt{2}+c\sqrt{5}+d\sqrt{10})=a+b\sqrt{2}-c\sqrt{5}-d\sqrt{10}$ for $a,b,c,d\in \mathbb{Q}$. 
\begin{enumerate}
\item  Verify $\tau$ is a field isomorphism.
\item  Prove $\sigma$ and $\tau$ generate a group inside the set of isomorphisms from $F$ to $F$. Provide a Cayley table or Cayley Diagram.
\end{enumerate}


\end{enumerate}

%DHW2Tue
\noindent Everyone will do the last question which examines maps from $\mathbb{Z}_{6 \cdot a}$ to $\mathbb{Z}_{6\cdot b}$. 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{packed_itemize}
\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 ($\mathbb{Z}_{6\cdot a} / \ker (\phi)$) isomorphic to?
\end{packed_itemize}



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