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Most development was happening on draft anyway, and master was lagging
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parents 252e9208 3605975a
## Algebra II
### Reading
***Note: I haven't got round to writing up and uploading all of my notes. This
should happen soon enough. See the*** `draft` ***branch for more up to date
lectures.***
See `lectures` directory for pre-compiled pdfs. See `draft` branch for latest
drafts; `master` branch is aimed at being stable, but might lag behind. I won't
be accepting PRs on the `draft` branch. However, I am currently rewriting the
notes in LaTeX because markdown became a pain. See `draft-latex` branch for
this.
### Reading
***Note: I haven't got round to writing up and uploading all of my notes. This
should happen soon enough.***
See `draft` branch for latest drafts; `master` branch is aimed at being stable,
but might lag behind. I won't be accepting PRs on the `draft` branch.
These are a set of lecture notes from Algebra II lectures held at the ETH
Zürich in the summer semester of 2019 by Prof. R. Pandharipande, written during
the semester.
Philip Weder has kindly submitted his summary of Algebra I to be included in
this project.
The proofs are intentionally written rather verbosely.
### Disclaimer
......@@ -31,20 +32,19 @@ open to any suggestions.
### Compiling yourself
These notes are compiled with and written in the `pandoc` markdown flavour. To
compile the pdfs yourself, install [pandoc](https://pandoc.org/) and call
TL;DR:
`latexmk -pdf *tex && latexmk -c`
`pandoc <lecture>.md -o <lecture>.pdf`
Previously, these notes were written in Markdown. As the code became more
complex, this became more difficult to maintain. Now they are written in LaTeX.
To compile the notes yourself, make sure you have LaTeX installed. Then call
to compile the whole set, call
`latexmk -pdf *tex`
`pandoc source/*md -o algebra-notes.pdf`
in the base directory of this project. To clean up all auxiliary files except
the `pdf` after compilation, call
Most *nix distros will have it available in the regular repositories. `pandoc`
is also available in OSX via [homebrew](https://brew.sh). For more help
installing see [the pandoc install guides](https://pandoc.org/installing.html).
There is also an [online](https://pandoc.org/try/) version that doesn't require
installing any tools.
`latexmk -c`
There are many more examples and demos on the `pandoc`
[website](https://pandoc.org/demos.html).
Most distributions of LaTeX will have this compilation tool by default.
% Algebra I Summary of Lecture Notes
% Summary from Summer Semester 2019, from the Lectures by Prof. R. Pandharipande
% written by Philipp Weder
\input{source/preamble}
\begin{document}
\input{source-I/title}
\input{source-I/preface}
\input{source-I/1-classical-number-theory.tex}
\input{source-I/2-excursus.tex}
\input{source-I/3-commutative-rings.tex}
\input{source-I/4-groups.tex}
\input{source-I/5-fields.tex}
\input{source-I/6-modules.tex}
\end{document}
% Algebra II Lecture Notes
% Notes from Summer Semester 2019, from the Lectures by Prof. R. Pandharipande
% written by Ryan Rueger
\input{source/preamble.tex}
\graphicspath{source-II/figures}
\renewcommand{\chaptername}{Lecture}
\newcommand{\incfig}[1]{%
\def\svgwidth{\textwidth /2}
\import{./source-II/figures/}{#1.pdf_tex}
}
\begin{document}
\input{source-II/title}
\input{source-II/preface}
\input{source-II/lecture-03-2019-03-08}
\input{source-II/lecture-04-2019-03-15}
\input{source-II/lecture-05-2019-03-22}
\input{source-II/lecture-06-2019-03-29}
\input{source-II/lecture-07-2019-04-05}
\input{source-II/lecture-08-2019-04-12}
\input{source-II/lecture-09-2019-05-03}
\input{source-II/lecture-11-2019-05-17}
\end{document}
\chapter{Classical Number Theory}
\section{Divisibility}
\begin{remark}
We assume that $\N$ satisfies the \textbf{Least Integer Axiom} (also
called the \emph{Well-Ordering Principle}): Every nonempty subset $C \subseteq
\N$ contains a smallest element; that is, there is $c_0 \in C$ with
$c_0 \leq c$.
\end{remark}
\begin{lemma}
If $a$ and $b$ are positive integers and $a | b$, then $a \leq b$.
\end{lemma}
\begin{theorem}[Division Algorithm]
If $a$ and $b$ are integers with $a \neq 0$, then there are unique integers
$q$ and $r$, called the \textbf{quotient} and \textbf{remainder}, with
%
\begin{align*}
b = qa + r \text{~and~} 0 \leq r < |a|
\end{align*}
\end{theorem}
\begin{definition}
A \textbf{common divisor} of integers $a$ and $b$ is an integer $c$ with $c |
a$ and $c | b$. The \textbf{greatest common divisor} of $a$ and $b$, denoted
by $\mathrm{gcd}(a,b)$, is defined by
%
\begin{align*}
\gcd(a,b) =
\begin{cases}
0 \text{~if~} a = 0 = b \\
\text{the largest common divisor of $a$ and $b$ otherwise}
\end{cases}
\end{align*}
\end{definition}
\begin{theorem}
If $a$ and $b$ are integers, then $\gcd(a,b)$ is a linear combination of $a$
and $b$.
\end{theorem}
\begin{corollary}
Let $a$ and $b$ be integers. A nonnegative common divisor $d$ is their $\gcd$
if and only if $c |d$ for every common divisor $c$ of $a$ and $b$.
\end{corollary}
\begin{definition}
An integer $p$ is \textbf{prime} if $p \geq 2$ and its only divisors are $\pm
1$ and $\pm p$. If an integer $a \geq 2$ is not prime, then it is called
\textbf{composite}.
\end{definition}
\begin{proposition}
Every integer $a \geq 2$ has a factorization
%
\begin{align*}
a = p_1 \dotsm p_t,
\end{align*}
%
where $p_1 \leq \ldots \leq p_t$ and all $p_i$ are prime.
\end{proposition}
\begin{definition}
If $a \geq 2$ is an integer, then a \textbf{prime factorization} of $a$ is
%
\begin{align*}
a = p_1^{e_1}p_2^{e_2} \hdots p_t^{e_t},
\end{align*}
%
where the $p_i$ are distinct primes and $e_i \geq 0$ for all $i$.
\end{definition}
\begin{theorem}[Euclid's Lemma]
If $p$ is a prime and $p \mid ab$, for integers $a$ and $b$, then $p \mid a$
or $p \mid b$. More generally, if $p \mid a_1 \hdots a_t$, then $p\mid a_i$
for some $i$.
Conversely, if $m \geq 2$ is an integer such that $m \mid ab$ always implies
$m \mid a$ or $m \mid b$, then $m$ is prime.
\end{theorem}
\begin{definition}
We call integers $a$ and $b$ \textbf{relatively prime} if their $\gcd$ is $1$.
\end{definition}
\begin{corollary}
Let $a,b$ and $c$ be integers. If $c$ and $a$ are relatively prime and if
$c\mid ab$, then $c\mid b$.
\end{corollary}
\begin{lemma}
Let $a$ and $b$ be integers.
%
\begin{enumerate}[(i)]
\item Then $\gcd(a,b) = 1$ if and only if $1$ is a linear combination
of $a$ and $b$.
\item If $d = \gcd(a,b)$ then the integers $a/d$ and $b/d$ are
relatively prime.
\end{enumerate}
\end{lemma}
\begin{theorem}[Fundamental Theorem of Arithmetic]
Every integer $a \geq 2$ has a unique factorization
%
\begin{align*}
a = p_1^{e_1} \hdots p_t^{e_t},
\end{align*}
%
where $p_1 < \ldots < p_t$, all $p_i$ are prime, and all $e_i > 0$.
\end{theorem}
\begin{corollary}
If $a = p_1^{e_1} \hdots p_t^{e_t}$ and $b = p_1^{f_1} \hdots p_t^{f_t}$ are
prime factorizations, then $a \mid b$ if and only if $e_i \leq f_i$ for all
$i$.
\end{corollary}
\begin{proposition}
Let $g$ and $h$ be divisors of $a$. If $\gcd(g,h) = 1$, then $gh\mid a$.
\end{proposition}
\begin{definition}
If $a,b$ are integers, then a \textbf{common multiple} is an integer $m$ with
$a \mid m$ and $b \mid m$. Their \textbf{least common multiple}, denoted by
%
\begin{align*}
\lcm(a,b),
\end{align*}
%
is their smallest common multiple. This definition extends in the obvious way
to give the $\lcm$ of integers $a_1, \ldots, a_n$.
\end{definition}
\begin{proposition}
If $a = p_1^{e_1} \hdots p_t^{e_t}$ and $b = p_1^{f_1} \hdots p_t^{f_t}$ are
prime factorizations, then
%
\begin{align*}
\gcd(a,b) = p_1^{m_1} \hdots p_t^{m_t} \text{~and~} \lcm(a,b)
= p_1^{M_1} \hdots p_t^{M_t},
\end{align*}
%
where $m_i = \min\{e_i, f_i\}$ and $M_i = \max\{e_i, f_i\}$.
\end{proposition}
\begin{corollary}
If $a$ and $b$ are integers, then
%
\begin{align*}
ab = \gcd(a,b) \lcm(a,b).
\end{align*}
\end{corollary}
\section{Euclidian Algorithm}
\begin{lemma}~
\begin{enumerate}
\item If $b = qa + r$, then $\gcd(a,b) = \gcd(r,a)$.
\item If $b \geq a$ are integers, then $\gcd(a,b) = \gcd(b-a,a)$.
\end{enumerate}
\end{lemma}
\section{Congruence}
\begin{definition}
Let $m\leq 0$ be fixed. Then integers $a$ and $b$ are \textbf{congruent
modulo} $m$, denoted by
%
\begin{align*}
a \equiv b \mod m,
\end{align*}
%
if $m \mid (a-b)$.
\end{definition}
\begin{proposition}
If $m\geq 0$ is a fixed integer, then for all integers $a,b,c$:
%
\begin{enumerate}
\item $a \equiv a \mod m;$
\item if $a \equiv b \mod m$, then $b \equiv a \mod m;$
\item if $a \equiv b \mod m$ and $ b \equiv c \mod m$, then $a \equiv c \mod
m$,
\end{enumerate}
%
i.e.\ congruence is a equivalence relation.
\end{proposition}
\begin{proposition}[elementary properties of congruence]
Let $m \geq 0$ be a fixed integer.
%
\begin{enumerate}
\item If $a = qm + r$, then $a \equiv r \mod m$.
\item If $0 \leq r' < r < m$, then $r \not \equiv r' \mod m$; that is, $r$
and $r'$ are not congruent $\mod m$.
\item $a \equiv b \mod m$ if and only if $a$ and $b$ leave the same
remainder after dividing by $m$.
\item If $m \geq 2$, each $a \in \Z$ is congruent $\mod m$ to exactly one of
$0,1,\ldots, m -1$.
\end{enumerate}
\end{proposition}
\begin{proposition}[Addition and multiplication]
Let $m \geq 0$ be a fixed integer.
\begin{enumerate}
\item If $a \equiv a' \mod m$ and $b \equiv b' \mod m$, then
%
\begin{align*}
a + b \equiv a' + b' \mod m.
\end{align*}
\item If $a \equiv a' \mod m$ and $b \equiv b' \mod m$, then
%
\begin{align*}
ab \equiv a'b' \mod m.
\end{align*}
\item If $a \equiv b \mod m$, then $a^n \equiv b^n \mod m$ for all $n
\geq 1$.
\end{enumerate}
\end{proposition}
\begin{proposition}~
\begin{enumerate}
\item If $p$ is prime, then $p \mid {p \choose r}$ for all $r$ with $0 < r <
p$, where $\binom p r$ is the binomial coefficient.
\item For integers $a$ and $b$,
%
\begin{align*}
(a+ b)^p \equiv a^p + b^p \mod p.
\end{align*}
\end{enumerate}
\end{proposition}
\begin{theorem}[Fermat]
If $p$ is prime, then
%
\begin{align*}
a^p \equiv a \mod p
\end{align*}
%
for every $a$ in $\Z$. More generally, for every integer $k \geq 1$,
%
\begin{align*}
a^{p^{k}} \equiv a \mod p.
\end{align*}
\end{theorem}
\begin{corollary}
If $p$ is prime and $m \equiv 1 \mod (p-1)$, then $a^m \equiv a \mod p$ for
all $a \in \Z$.
\end{corollary}
\begin{theorem}
If $\gcd(a,m) = 1$, then, for every integer $b$, the congruence
%
\begin{align*}
ax \equiv b \mod m
\end{align*}
%
can be solved for $x$; in fact, $x = sb$, where $sa \equiv 1 \mod m$, is one
solution. Moreover, any two solutions are congruent $\mod m$.
\end{theorem}
\begin{theorem}[Chinese Remainder Theorem]
If $m$ and $m'$ are relatively prime, then the two congruences
%
\begin{align*}
x & \equiv b \mod m \\
x & \equiv b' \mod m' \\
\end{align*}
%
have a common solution, and any two solutions are congruent $\mod mm'$.
\end{theorem}
\chapter{Excursus to Zorn's Lemma and Categories}
\section{The Axiom of Choice}
Let $I$ be an index set. For every $i \in I$, let $X_i$ be a nonempty set. Then
%
\begin{align*}
\prod_{i \in I}X_i
\end{align*}
%
is nonempty, i.e.\ there exists $f: I \to \bigcup_{i \in I} X_i$, such
that $f(i) \in X_i$ for each $i \in I$.
\paragraph{Objection.} If $X_i \subset X$, then $f: I \to X$ chooses some
element $x_i = f(i)$ of $X_i$ for each $i$. How is this choice made?
\begin{theorem}
This cannot be deduced from the other axioms of set theory if it is consistent.
\end{theorem}
\begin{definition}
Let $X$ be a set. An \textbf{ordering} of $X$ is a relation $\cR$ on
$X$, such that
%
\begin{enumerate}
\item $x \cR x$
\item $(x\cR y \text{ and } y \cR x) \implies x = y$
\item $(x \cR y \text{ and } y \cR z) \implies x \cR z$.
\end{enumerate}
\end{definition}
\begin{definition}
Let $X$ be a set and $\cR$ an ordering on $X$. $x \in X$ is called a
maximal element of $X$, if $x \cR y \implies y = x$.
\end{definition}
\begin{definition}
An order $\cR$ on $X$ is \textbf{total} if for all $x,y \in X$ either $x \cR
y$ or $y \cR x$.
\end{definition}
\begin{remark}
For a maximal element $x \in X$ with a relation $\cR$ it does not always hold
that $y \cR x$ for all $y \in X$. Furthermore, a maximal element is not
necessarily unique.
\end{remark}
\begin{definition}
A set $X$ is \textbf{Partially ordered} (a poset) if there is a relation $x
\preceq y$ defined on $X$ which is
%
\begin{enumerate}
\item \textbf{Reflexive:} $x \preceq x$ for all $x \in X$;
\item \textbf{Anti-Symmetric:} if $x \preceq y$ and $y \preceq x$, then $x =
y$;
\item \textbf{Transitive:} if $x \preceq y$ and $y \preceq z$, then $x
\preceq z$.
\end{enumerate}
\end{definition}
\begin{definition}
A poset $X$ is a \textbf{chain} (or is \textbf{simply ordered} or is
\textbf{totally ordered}) if, for all $x,y \in X$, either $x \preceq y$ or $y
\preceq x$. An \textbf{upper bound} of a nonempty subset $Y$ of a poset $X$ is
an element $x_0 \in X$, not necessarily in $Y$,with $y \preceq x_0$ for every
$y \in Y$.
\end{definition}
\begin{theorem}[Zorn's Lemma]
If $X$ is a nonempty poset in which every chain has an upper bound in $X$,
then $X$ has a maximal element.
\end{theorem}
\begin{lemma}
If $C$ is a chain in a poset $X$ and $S = \{c_1, \dotsc, c_n\}$ is a finite
subset of $C$, then there exists some $c_i$ with $c_j \preceq c_i$ for all
$c_j \in S$.
\end{lemma}
\begin{theorem}
The Axiom of Choice and Zorn's Lemma are equivalent.
\end{theorem}
\begin{theorem}
Let $K$ be a field. Let $V$ be a $K$-vector space. Then $V$ has a basis.
\end{theorem}
\section{The Language of Categories and Functors}
\begin{definition}
A category $\mathcal{C}$ consists of three ingredients:
%
\begin{enumerate}
\item A class $\Obj(\mathcal{C})$ of \textbf{objects},
\item A set of \textbf{morphisms} (or \textbf{arrows}) $\Hom(A,B)$ for every
ordered pair $(A,B)$ of objects
\item A \textbf{composition} $\Hom(A,B) \times \Hom(B,C) \to
\Hom(A,C)$ denoted by
%
\begin{align*}
(f,g) \mapsto gf,
\end{align*}
%
for every ordered triple $(A,B,C)$ of objects.
\end{enumerate}
%
These ingredients have to satisfy the following axioms:
%
\begin{enumerate}
\item $\Hom$ sets are pairwise disjoint; that is, each morphism $f \in
\Hom(A,B)$ has a unique \textbf{domain} $A$ and unique \textbf{target}
$B$.
\item For each object $A$, there is an \textbf{identity morphism} $\id_A \in
\Hom(A,A)$ such that
%
\begin{align*}
\forall f: A \to B: f \id_A = f \text{~and~} \id_B f = f
\end{align*}
\item Composition is associative: given morphisms
%
\begin{align*}
A \xrightarrow{f} B \xrightarrow{g} C \xrightarrow{h} D,
\end{align*}
%
we have
%
\begin{align*}
h(gf) = (hg)f.
\end{align*}
\end{enumerate}
\end{definition}
\begin{definition}
Let $\mathcal{C}$ be a category. $f: A \to B$ (in $\mathcal{C}$) is an
\textbf{isomorphism} if there exists $g: B \to A$ in $\mathcal{C}$
such that
%
\begin{align*}
gf = \id_A \text{~and~} fg = \id_B.
\end{align*}
%
$g$ is called the \textbf{inverse} of $f$.
\end{definition}
\begin{proposition}~
\begin{enumerate}
\item $g$ is unique if it exists.
\item $1_{A}$ is an isomorphism with $\id_{A}^{-1} = \id_{A}$.
\item If $f$ and $g$ are isomorphisms, then $f \circ g$ and $(f \circ
g)^{-1} = g^{-1} \circ f^{-1}$ are also isomorphisms.
\end{enumerate}
\end{proposition}
\begin{definition}
If $\mathcal{C}$ and $\mathcal{D}$ are categories, then a \textbf{functor} $T:
\mathcal{C} \to \mathcal{D}$ is a function such that
%
\begin{enumerate}
\item if $A \in \Obj(\mathcal{C})$, then $T(A) \in \Obj(\mathcal{D})$;
\item if $f: A \to A'$ in $\mathcal{C}$, then $T(f): T(A)
\to T(A')$ in $\mathcal{D}$;
\item if $A \overset{f}{\to} A' \overset{g}{\to} A''$ in $\mathcal{C}$, then
$T(A) \overset{T(f)}{\to} T(A') \overset{T(g)}{\to} T(A'')$ in
$\mathcal{D}$ and
%
\begin{align*}
T(gf) = T(g) T(f);
\end{align*}
\item for every $A \in \Obj(\mathcal{C})$,
%
\begin{align*}
T(1_{A}) = \id_{T(A)}
\end{align*}
\end{enumerate}
\end{definition}
This diff is collapsed.
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\chapter{Fields}
\section{Field Extensions and Algebraic Elements}
\begin{proposition}
If $k$ is a field and $I = (f)$, where $f(x)$ is a nonzero polynomial in
$k[x]$, then the following are equivalent:
\begin{enumerate}
\item $f$ is irreducible;
\item $k[x]/I$ is a field;
\item $k[x]/I$ is a domain.
\end{enumerate}