\documentclass[numbers=enddot,12pt,final,onecolumn,notitlepage]{scrartcl}%
\usepackage[headsepline,footsepline,manualmark]{scrlayer-scrpage}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsthm}
\usepackage{color}
\usepackage{hyperref}
\usepackage[sc]{mathpazo}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{enumitem}
\usepackage{graphicx}%
\setcounter{MaxMatrixCols}{30}
%TCIDATA{OutputFilter=latex2.dll}
%TCIDATA{Version=5.50.0.2960}
%TCIDATA{LastRevised=Tuesday, July 28, 2026 22:49:56}
%TCIDATA{}
%TCIDATA{}
%TCIDATA{BibliographyScheme=Manual}
%BeginMSIPreambleData
\providecommand{\U}[1]{\protect\rule{.1in}{.1in}}
%EndMSIPreambleData
\setlength\textheight{22.5cm}
\setlength\textwidth{15cm}
\setlength\oddsidemargin{0.5cm}
\setlength\evensidemargin{0.5cm}
\setlength\topmargin{-1cm}
\clearpairofpagestyles
\ihead{Errata to \textit{Summer Term Abstract Algebra}}
\ohead{July 28, 2026}
\cfoot{\pagemark}
\newcommand{\ZZ}{\mathbb Z}
\newcommand{\RR}{\mathbb R}
\newcommand{\CC}{\mathbb C}
\newcommand{\FF}{\mathbb F}
\newcommand{\Orb}{\operatorname{Orb}}
\begin{document}
\begin{center}
\textbf{Summer Term Abstract Algebra}
\textit{Samir Siksek}
Handouts I--IV; 55 pages in total
\url{https://samirsiksek.github.io/siksek.github.io/index.html}
\textbf{Errata and comments} by Darij Grinberg
\end{center}
\noindent The page numbers below refer to the printed page numbers in each
handout; thus the pagination restarts at page~1 four times. The four supplied
PDFs are dated April~30, April~28, May~18, and May~28, 2020, respectively.
This list is not claimed to be exhaustive. It concentrates on mathematical
errors, proof gaps, and points likely to cause avoidable difficulty for a
student. The corrections to references into \textit{Introduction to Abstract
Algebra} use the version dated September~22, 2020.
Most of the errata below were found with assistance from GPT-5.6 Sol. All have
been verified by myself.
\appendix
\section{Handout I: Finite Fields}
\begin{enumerate}
\item \textbf{Page 1, opening of Section 2:} In the September~22, 2020 version
of \textit{Introduction to Abstract Algebra}, rings and fields are Chapters
XIV and XV, not Chapters XV and XVI.
\item \textbf{Page 2, opening list of facts:} The assertion \textquotedblleft%
$\mathbb{Z}/m\mathbb{Z}$ is a field if and only if $m$ is
prime\textquotedblright\ needs the usual hypothesis $m\geq2$ (or, at least,
that $m$ is a positive integer).
\item \textbf{Page 3, Theorem 2:} The quotient $q$ and remainder $r$ are
\emph{unique}, as in the real-polynomial division theorem recalled just above.
Thus replace \textquotedblleft Then there are $q,r\in K[X]$
with\textquotedblright\ by \textquotedblleft Then there are unique $q,r\in
K[X]$ with\textquotedblright.
\item \textbf{Page 5, opening of Section 4:} In the September~22, 2020 version
of \textit{Introduction to Abstract Algebra}, quotients of additive abelian
groups are treated in Chapter XII, not Chapter XIII.
\item \textbf{Page 6, Exercise 5:} For the same version of
\textit{Introduction to Abstract Algebra}, the cited result is Lemma XII.17,
not Lemma XIII.17.
\item \textbf{Page 9, proof of Theorem 12(c):} After writing $f=f_{1}f_{2}$,
the literal equality \textquotedblleft$\gcd(f,f_{1})=f_{1}$\textquotedblright%
\ requires $f_{1}$ to have been chosen monic, since polynomial gcds are being
normalized to be monic. It is enough to say that $\gcd(f,f_{1})$ has positive
degree, and hence is not $1$.
At the end of the proof, \textquotedblleft composite\textquotedblright\ should
be \textquotedblleft reducible\textquotedblright.
\item \textbf{Page 10:} Again, \textquotedblleft composite\textquotedblright%
\ (in \textquotedblleft The first three are composite\textquotedblright)
should be \textquotedblleft reducible\textquotedblright.
\item \textbf{Page 12, reduction of $\theta^{n+1}$:} In the long displayed
computation, replace \textquotedblleft$-a_{1}\theta-a_{1}\theta^{2}%
-\cdots--a_{n-2}\theta^{n-1}$\textquotedblright\ by \textquotedblleft%
$-a_{0}\theta-a_{1}\theta^{2}-\cdots-a_{n-2}\theta^{n-1}$\textquotedblright.
\end{enumerate}
\section{Handout II: Cosets and Lagrange's Theorem}
\begin{enumerate}
\item \textbf{Page 8, Example 15:} In the September~22, 2020 MA136 notes, the
parity result is Theorem XIII.40, not Theorem XIV.40.
\item \textbf{Page 11, opening of Section 10:} Replace \textquotedblleft We
checked that $\operatorname{GL}_{2}(K)$ is a group and $\operatorname{SL}%
_{n}(K)$ is a subgroup\textquotedblright\ by \textquotedblleft We checked that
$\operatorname{GL}_{n}(K)$ is a group and $\operatorname{SL}_{n}(K)$ is a
subgroup\textquotedblright.
\end{enumerate}
\section{Handout III: Cyclic and Dihedral Groups}
\begin{enumerate}
\item \textbf{Pages 8--9, derivation of the rotation matrix:} The polar angle
$\phi$ is not determined when the vector $\mathbf{x}$ is zero. After
introducing the polar coordinates $(r,\phi)$, add that $\phi$ may be chosen
arbitrarily when $r=0$ (or handle the zero vector separately).
\item \textbf{Page 11, opening of Section 6:} In the September~22, 2020
version of \textit{Introduction to Abstract Algebra}, the notation for the
symmetries of the square is in Section IV.4, not Section V.4.
\item \textbf{Page 13, display (4):} Delete the extra closing brace after
\textquotedblleft$r^{n-1}s$\textquotedblright.
\end{enumerate}
\section{Handout IV: Symmetric Groups}
\begin{enumerate}
\item \textbf{Page 1, opening of Section 2:} In the September~22, 2020 version
of \textit{Introduction to Abstract Algebra}, symmetric groups are Chapter
XIII, not Chapter XIV.
\item \textbf{Page 7, definition of orbit:} It is worth pointing out that
$\operatorname*{Orb}\nolimits_{\rho}\left( a\right) $ can also be written
as
\[
\operatorname*{Orb}\nolimits_{\rho}\left( a\right) =\left\{ a,\rho\left(
a\right) ,\rho^{2}\left( a\right) ,\ldots\right\} ,
\]
because the equality $\rho^{u}\left( a\right) =a$ entails that the sequence
$\left( a,\rho\left( a\right) ,\rho^{2}\left( a\right) ,\ldots\right) $
is periodic with period $u$. This equality is used in the proof of Lemma 12
further below (e.g., when you derive $b\in\operatorname*{Orb}\nolimits_{\rho
}\left( a\right) $ from $b=\rho^{k}\left( a\right) $, and likewise when
you derive $\rho^{k+t}\left( a\right) \in\operatorname*{Orb}\nolimits_{\rho
}\left( a\right) $).
\item \textbf{Page 8, proof of Theorem 8:} Replace \textquotedblleft%
$\{1,2,\ldots,n\}=C_{1}\cup C_{2}\cup\cdots\cup C_{n}$\textquotedblright\ by
\textquotedblleft$\{1,2,\ldots,n\}=C_{1}\cup C_{2}\cup\cdots\cup C_{k}%
$\textquotedblright.
\item \textbf{Pages 12--13, Theorem 19 and Corollary 20:} These are only true
if $n\geq2$. For $n=1$, one has $A_{1}=S_{1}$, so $[S_{1}:A_{1}]=1$ and
$\#A_{1}=1$, not $1!/2$.
\item \textbf{Page 15, first permutation-type example:} Replace
\textquotedblleft has permutation type $[4,2,1,0,0,0,0,0,0,0,0,0]$%
\textquotedblright\ by \textquotedblleft has permutation type
$[4,2,1,0,0,0,0,0,0,0,0]$\textquotedblright\ (the tuple must have $11$
entries, not $12$, so it had one $0$ too much at the end).
\item \textbf{Page 15, equation (9):} Insert the missing plus sign after
$3\alpha_{3}$. The equation should read
\[
\alpha_{1}+2\alpha_{2}+3\alpha_{3}+\cdots+n\alpha_{n}=n.
\]
\end{enumerate}
\end{document}