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\usetikzlibrary{lindenmayersystems} % This is for drawing Sierpinski triangles.

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%	VARIOUS USEFUL COMMANDS
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% The commands below might be convenient. For example, you probably
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% of $V$, rather than writing $\mathcal{P}_2(V)$.
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% they occur in subscripts or superscripts. So you should not write
% $K_\CC$, but rather $K_{\CC}$.
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%	METADATA
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\newcommand{\myname}{Darij Grinberg} % ENTER YOUR NAME HERE
\newcommand{\mymail}{darij.grinberg@drexel.edu} % ENTER YOUR EMAIL HERE
\newcommand{\psetnumber}{1} % ENTER THE NUMBER OF THIS PSET HERE

%----------------------------------------------------------------------------------------
%	HEADER AND FOOTER
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\ihead{Solutions to homework set \#\psetnumber} % Page header left
\ohead{page \thepage\ of \pageref{LastPage}} % Page header right
\ifoot{\myname} % left footer
\ofoot{\mymail} % right footer

%----------------------------------------------------------------------------------------
%	TITLE SECTION
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\title{	
\normalfont \normalsize 
\textsc{Drexel University, Department of Mathematics} \\ [15pt] % Your university, school and/or department name(s)
\horrule{0.5pt} \\[0.4cm] % Thin top horizontal rule
\huge Math 332: Undergraduate Abstract Algebra II, \\
Winter 2025:
Homework \psetnumber\\% The assignment title
\horrule{2pt} \\[0.5cm] % Thick bottom horizontal rule
Please solve \textbf{at most 3 of the 6 problems!}\footnote{%
I recommend solving as many problems as you can and wish,
but I will only grade and score 3 solutions per submission
(and if you submit more, I get to pick which ones I grade).
\\
Results stated in class, and claims of previous problems
(even if you did not solve these previous problems), can
be used without proof. For example, in solving Problem 5,
you can use the result of Problem 2 without proof.
\\
I expect approximately the level of detail that I give in class.
Purely straightforward arguments (like checking the ring
axioms for a direct product of rings) need not be spelled out;
I only expect a note of their necessity.
}
\horrule{2pt} \\[0.5cm] % Thick bottom horizontal rule
}
\author{\myname}

\begin{document}

\maketitle % This command causes the title part to be printed.

%----------------------------------------------------------------------------------------
%	EXERCISE 1
%----------------------------------------------------------------------------------------
\horrule{0.3pt} \\[0.4cm]

\section{Exercise 1}

\subsection{Problem}

\begin{enumerate}

\item[\textbf{(a)}] Without computing the integer $7^4$,
prove that $\overline{7}^4 = \overline{1}$ in the ring
$\ZZ / 10$.

\item[\textbf{(b)}] Find a simple rule for the $k$-th
power $\overline{7}^k$ of the element $\overline{7}$ in
the ring $\ZZ / 10$. Specifically, this rule should
express $\overline{7}^k$ in terms of the remainder that
$k$ leaves when divided by $4$.

\item[\textbf{(c)}] What is the units digit of the
number $7^{9999}$ ?

\end{enumerate}

\subsection{Solution}

...

%----------------------------------------------------------------------------------------
%	EXERCISE 2
%----------------------------------------------------------------------------------------
\horrule{0.3pt} \\[0.4cm]

\section{Exercise 2}

\subsection{Problem}

\begin{enumerate}

\item[\textbf{(a)}] Prove that every element
$x \in \ZZ / 5$ satisfies $x^5 = x$ in $\ZZ / 5$.

\item[\textbf{(b)}] In the ring $\mathbb{H}$ of
Hamilton quaternions (defined in \S 2.1.2 of the
notes), compute $ijk$ and $\tup{1 + i + j + k}^2$.

\end{enumerate}

Next, recall the ring $F_4$ constructed in \S 2.1.2,
with its four elements $0, 1, a, b$.

\begin{enumerate}

\item[\textbf{(c)}] Prove that $a^4 = a$ in this ring.

\item[\textbf{(d)}] What is $b^4$ ?

\end{enumerate}

\subsection{Solution}

...

%----------------------------------------------------------------------------------------
%	EXERCISE 3
%----------------------------------------------------------------------------------------
\horrule{0.3pt} \\[0.4cm]

\section{Exercise 3}

\subsection{Problem}

Let $p$ be a prime number, and let $k$ be a positive integer.

\begin{enumerate}

\item[\textbf{(a)}] Prove that the only
idempotent\footnote{An element
$a$ of a ring is said to be \emph{idempotent}
if $a^2 = a$.} elements of the ring $\ZZ / p^k$
are $\overline{0}$ and $\overline{1}$.

\item[\textbf{(b)}] Now assume furthermore that
$p \neq 2$. Prove that the only
involutive\footnote{An element
$a$ of a ring is said to be \emph{involutive}
if $a^2 = 1$.} elements
of the ring $\ZZ / p^k$ are $\overline{1}$ and
$\overline{-1}$.

\end{enumerate}

\subsection{Solution}

...

%----------------------------------------------------------------------------------------
%	EXERCISE 4
%----------------------------------------------------------------------------------------
\horrule{0.3pt} \\[0.4cm]

\section{Exercise 4}

\subsection{Problem}

Let $R$ be a ring. Prove the following:

\begin{enumerate}

\item[\textbf{(a)}]
If $a$ is an idempotent element of $R$, then
$1-a \in R$ is again idempotent.

\item[\textbf{(b)}]
If $a$ is an involutive element of $R$, then
$-a \in R$ is again involutive.

\item[\textbf{(c)}]
If $a$ is an idempotent element of $R$, then
$a^n = a$ for each positive $n \in \NN$.

\item[\textbf{(d)}]
If $a$ is an idempotent element of $R$, then
$\tup{1+a}^n = 1 + \tup{2^n - 1} a$ for each
$n \in \NN$.

\end{enumerate}

\subsection{Solution}

...

%----------------------------------------------------------------------------------------
%	EXERCISE 5
%----------------------------------------------------------------------------------------
\horrule{0.3pt} \\[0.4cm]

\section{Exercise 5}

\subsection{Problem}

There is a certain ring $F_8$ consisting of eight
distinct elements $0, 1, a, b, c, d, e, f$.
Its addition $+$ and its multiplication $\cdot$
are given by the following tables:
\begin{align*}
&
\begin{tabular}
[c]{|c||c|c|c|c|c|c|c|c|}\hline
$x+y$ & $y=0$ & $y=1$ & $y=a$ & $y=b$ & $y=c$ & $y=d$ & $y=e$ & $y=f$%
\\\hline\hline
$x=0$ & $0$ &  &  &  &  &  &  & \\\hline
$x=1$ &  & $0$ &  &  &  &  &  & \\\hline
$x=a$ &  & $b$ & $0$ &  &  &  &  & \\\hline
$x=b$ &  & $a$ & $1$ &  &  &  &  & \\\hline
$x=c$ &  & $d$ & $e$ &  & $0$ &  &  & \\\hline
$x=d$ &  & $c$ & $f$ &  & $1$ &  &  & \\\hline
$x=e$ &  & $f$ & $c$ &  & $a$ &  &  & \\\hline
$x=f$ &  & $e$ & $d$ &  & $b$ &  &  & \\\hline
\end{tabular}
\\
& \\
&
\begin{tabular}
[c]{|c||c|c|c|c|c|c|c|c|}\hline
$x\cdot y$ & $y=0$ & $y=1$ & $y=a$ & $y=b$ & $y=c$ & $y=d$ & $y=e$ &
$y=f$\\\hline\hline
$x=0$ &  &  &  &  &  &  &  & \\\hline
$x=1$ &  & $1$ &  &  &  &  &  & \\\hline
$x=a$ &  & $a$ & $c$ &  & $b$ &  &  & \\\hline
$x=b$ &  & $b$ &  &  &  &  &  & \\\hline
$x=c$ &  & $c$ & $b$ &  & $e$ &  &  & \\\hline
$x=d$ &  &  &  &  &  &  &  & \\\hline
$x=e$ &  &  &  &  &  &  &  & \\\hline
$x=f$ &  &  &  &  &  &  &  & \\\hline
\end{tabular}
\qquad .
\end{align*}
Oops, I lost most of the entries! Reconstruct all
missing entries in the tables. (You can take it
for granted that $F_{8}$ really is a ring with
zero $0$ and unity $1$.
Proofs are not required in this problem.)

\subsection{Solution}

...

%----------------------------------------------------------------------------------------
%	EXERCISE 6
%----------------------------------------------------------------------------------------
\horrule{0.3pt} \\[0.4cm]

\section{Exercise 6}

\subsection{Problem}

Let $n \in \NN$. Let $R$ be any ring.
An $n \times n$-matrix
$A = \begin{pmatrix}
a_{1,1} & a_{1,2} & \cdots & a_{1,n} \\
a_{2,1} & a_{2,2} & \cdots & a_{2,n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{n,1} & a_{n,2} & \cdots & a_{n,n}
\end{pmatrix} \in R^{n \times n}$ will be called
\emph{centrosymmetric} if it satisfies
\[
a_{i,j} = a_{n+1-i,\ n+1-j}
\qquad \text{for all } i,j \in \set{1, 2, \ldots, n} .
\]
(Visually, this means that $A$ is preserved under
``$180^{\circ}$-rotation'', i.e., that any two cells
of $A$ that are mutually symmetric across the center
of the matrix have the same entry. For example, a
centrosymmetric $4\times4$-matrix has the form
$\begin{pmatrix}
a & b & c & d \\
e & f & g & h \\
h & g & f & e \\
d & c & b & a
\end{pmatrix}$ for $a, b, \ldots, h \in R$.)

Prove that the set $\set{\text{all centrosymmetric }
n \times n\text{-matrices with entries in }R}$ is a
subring of $R^{n \times n}$.

\subsection{Hint}

This can be done in a particularly
slick way as follows: Let $W$ be the
$n \times n$-matrix obtained from the identity
matrix $I_n$ by a horizontal reflection (or,
equivalently, a vertical reflection). For example,
if $n = 4$, then $W = \begin{pmatrix}
0 & 0 & 0 & 1 \\
0 & 0 & 1 & 0 \\
0 & 1 & 0 & 0 \\
1 & 0 & 0 & 0
\end{pmatrix}$.
Now, show that an $n \times n$-matrix $A$ is
centrosymmetric if and only if it satisfies
$AW = WA$.

\subsection{Solution}

...


\horrule{0.3pt} \\[0.4cm]

% If you want to cite some sources, uncomment the following
% lines (i.e., remove the "%" symbols at the start of each
% line) and adapt it accordingly.

% \begin{thebibliography}{99999999}                                                                                         %

% \bibitem[DF]{DF}
% David S. Dummit, Richard M. Foote,
% \textit{Abstract Algebra},
% 3rd edition,
% Wiley 2004.

% \bibitem[23wa]{notes}
% Darij Grinberg,
% \textit{Math 332 Winter 2023 notes: An introduction
% to the algebra of rings and fields},
% 12 January 2023.
% \url{https://www.cip.ifi.lmu.de/~grinberg/t/23wa/23wa.pdf}

% \end{thebibliography}

\end{document}

