update

10 files changed, 182 insertions, 26 deletions

diff --git a/semestr2/algebra2r/notatki/notatki.pdf b/semestr2/algebra2r/notatki/notatki.pdf
index 9c9aec4..15ecfe1 100644
--- a/semestr2/algebra2r/notatki/notatki.pdf
+++ b/semestr2/algebra2r/notatki/notatki.pdf
diff --git a/semestr2/algebra2r/notatki/notatki.tex b/semestr2/algebra2r/notatki/notatki.tex
index ee16bfa..f715517 100644
--- a/semestr2/algebra2r/notatki/notatki.tex
+++ b/semestr2/algebra2r/notatki/notatki.tex
@@ -1,5 +1,4 @@
-\documentclass[11pt, a4paper, final]{amsart} 
-\setlength{\emergencystretch}{2em}
+\documentclass[11pt, a4paper, final]{amsart} \setlength{\emergencystretch}{2em}
 
 \usepackage[utf8]{inputenc}
 
@@ -54,9 +53,13 @@
 \DeclareMathOperator{\Gal}{Gal} 
 \DeclareMathOperator{\Int}{{Int}}
 \DeclareMathOperator{\ord}{{ord}}
+\DeclareMathOperator{\acl}{{acl}}
 \DeclareMathOperator{\Tr}{{Tr}}
 \DeclareMathOperator{\rng}{{Rng}}
 \DeclareMathOperator{\dom}{{Dom}}
+\DeclareMathOperator{\alg}{{alg}}
+\DeclareMathOperator{\End}{{End}}
+\DeclareMathOperator{\trdeg}{{trdeg}}
 
 \newcommand{\cupdot}{\mathbin{\mathaccent\cdot\cup}} 
 \newcommand{\cC}{\mathcal C} 
@@ -64,6 +67,7 @@
 \newcommand{\cV}{\mathcal{V}} 
 \newcommand{\cU}{\mathcal{U}}
 \newcommand{\cG}{\mathcal{G}}
+\newcommand{\cP}{\mathcal{P}}
 \newcommand{\cH}{\mathcal{H}}
 \newcommand{\bN}{\mathbb N} 
 \newcommand{\bR}{\mathbb R}
@@ -74,6 +78,7 @@
 \newcommand{\cK}{\mathcal K}
 \newcommand{\cL}{\mathcal L}
 \newcommand{\cT}{\mathcal T}
+\newcommand{\cN}{\mathcal N}
 \newcommand{\defeq}{\overset{\text{def}}{=}}
 
 \newcommand{\FrAut}{\Pi}
@@ -143,22 +148,29 @@
 \newcommand{\xqed}[1]{% \leavevmode\unskip\penalty9999 \hbox{}\nobreak\hfill
   \quad\hbox{\ensuremath{#1}}}
 
-\title{Tytuł} 
-\author{Franciszek Malinka}
+% \usepackage{blindtext}
+% \usepackage{titlesec}
+
+\title{Algebra 2R notes} 
+\author{Franciszek Malinka\\University of Wrocław}
+\date{June 2023}
+
 
 \begin{document}
-  \newpage
-  \thispagestyle{empty}
-  \begin{center}
-	\textbf{\textit{\large Franciszek Malinka}}\\
-	\vspace{0.5cm}
-	{\Large Algebra 2R notes}\\
-  \end{center}
-  \vspace{3cm}
-  \vfill
-  \begin{center}
-	{\large Wrocław 2023}\\
-  \end{center}
+ %  \newpage
+ %  \thispagestyle{empty}
+ %  \begin{center}
+	% \textbf{\textit{\large Franciszek Malinka}}\\
+	% \vspace{0.5cm}
+	% {\Large Algebra 2R notes}\\
+ %  \end{center}
+ %  \vspace{3cm}
+ %  \vfill
+ %  \begin{center}
+	% {\large Wrocław 2023}\\
+ %  \end{center}
+
+  \maketitle
 
   \newpage
 
@@ -186,6 +198,8 @@
   \href{franciszek.malinka@gmail.com}{write me an email} or 
   \href{https://www.buymeacoffee.com/framal}{buy me a coffee}.
 
+  \tableofcontents
+
   \section{Preliminaries} 
   \label{section:preliminaries}
   \subfile{sections/prelim}
@@ -203,7 +217,7 @@
   \label{section:finite-fields}
   \subfile{sections/finite-fields}
 
-  \section{Galois extensions}
+  \section{Galois group}
   \label{section:galois}
   \subfile{sections/galois}
 
@@ -218,4 +232,8 @@
   \section{Transcendental extensions}
   \label{section:lioville}
   \subfile{sections/lioville}
+
+  \section{Modules}
+  \label{section:modules}
+  \subfile{sections/modules}
 \end{document}
diff --git a/semestr2/algebra2r/notatki/sections/alg-ext.tex b/semestr2/algebra2r/notatki/sections/alg-ext.tex
index 4e9ae10..179bb3a 100644
--- a/semestr2/algebra2r/notatki/sections/alg-ext.tex
+++ b/semestr2/algebra2r/notatki/sections/alg-ext.tex
@@ -1,4 +1,4 @@
-\documentclass[../notes.tex]{subfiles}
+\documentclass[../notatki.tex]{subfiles}
 \begin{document}
   \subsection{Algebraic closeness}
   \begin{definition}
diff --git a/semestr2/algebra2r/notatki/sections/finite-fields.tex b/semestr2/algebra2r/notatki/sections/finite-fields.tex
index 4e623f0..2f68540 100644
--- a/semestr2/algebra2r/notatki/sections/finite-fields.tex
+++ b/semestr2/algebra2r/notatki/sections/finite-fields.tex
@@ -1,8 +1,9 @@
-\documentclass[../notes.tex]{subfiles}
+\documentclass[../notatki.tex]{subfiles}
 \begin{document}
   Non-trivial finite fields can be only of $\Char = p$ prime. Also, we can
   describe all of them very specifically.
 
+  \subsection{Basic facts}
   \begin{remark}
     Let $K$ be a finite field $\Char = p > 0$, Then $K^{*}$ (the multiplicative
     subgroup) is also finite and for $n = |K|$ we have $x^n = x$ for any $x\in
diff --git a/semestr2/algebra2r/notatki/sections/galois-cont.tex b/semestr2/algebra2r/notatki/sections/galois-cont.tex
index 119e31a..2896ce9 100644
--- a/semestr2/algebra2r/notatki/sections/galois-cont.tex
+++ b/semestr2/algebra2r/notatki/sections/galois-cont.tex
@@ -1,4 +1,4 @@
-\documentclass[../notes.tex]{subfiles}
+\documentclass[../notatki.tex]{subfiles}
 \begin{document}
   \subsection{Galois extension and Artin's theorem}
   So far we were focusing on the absolute Galois group $G(\hat{K}/K)$. In this
diff --git a/semestr2/algebra2r/notatki/sections/galois.tex b/semestr2/algebra2r/notatki/sections/galois.tex
index 5adefb2..e15aa10 100644
--- a/semestr2/algebra2r/notatki/sections/galois.tex
+++ b/semestr2/algebra2r/notatki/sections/galois.tex
@@ -1,6 +1,6 @@
-\documentclass[../notes.tex]{subfiles}
+\documentclass[../notatki.tex]{subfiles}
 \begin{document}
-  \subsection{Galois group}
+  \subsection{Definition}
 
   From now on we work with algebraic extensions $K\subset L\subset M$ and we will assume 
   that $L\subset M\subset \hat{K}$.
diff --git a/semestr2/algebra2r/notatki/sections/lioville.tex b/semestr2/algebra2r/notatki/sections/lioville.tex
index 680bed1..ecfb0b9 100644
--- a/semestr2/algebra2r/notatki/sections/lioville.tex
+++ b/semestr2/algebra2r/notatki/sections/lioville.tex
@@ -1,4 +1,4 @@
-\documentclass[../notes.tex]{subfiles}
+\documentclass[../notatki.tex]{subfiles}
 \begin{document}
   \subsection{Lioville numbers}
   \begin{lemma}
@@ -12,4 +12,141 @@
   not algebraic, e.g. that $\sum_{n=1}^\infty \frac{1}{2^{n!}}$ is
   transcendental.
 
+  \subsection{Transcendental extensions}
+
+  \begin{definition}
+    The field extension $K\subset L$ is \emph{transcendental} when there is an alement
+    $a\in L$ transcendental over $K$. We say that $K\subset L$ is \emph{purely
+    transcendental} when $\forall a\in L\setminus K \; a/K$ is transcendental.
+  \end{definition}
+
+  \begin{remark}
+    For $a\in L\supset K$, $a/K$ is transcendental $\Leftrightarrow$ $K(a)\cong K(X)$.
+  \end{remark}
+
+  This remark is quite intuitive: if $K$ cannot say anything about $a$, then it
+  is as esoteric as $X$ from its perspective. I think this remark gives a good 
+  framework to work with transcendental extensions and is useful in proofs.
+
+  In the further part of this note we'll give definitions of transcendence
+  basis and algebraic independence, but in an unusual way, as they'll come from
+  model theory and the notion of the algebraic closure.
+
+  From now on imagine that there is a huge algebraically closed field $\cU$, it
+  is bigger than any of the fields in any definition or theorem we'll describe
+  in a moment and we assume that all mentioned fields are contained inside
+  $\cU$ (see \href{https://modeltheory.fandom.com/wiki/Monster_model}{this post
+  on modeltheory.fandom.com}).
+
+  \begin{definition}
+    Let $K\subset \cU$ be a field and $F\subset K$ be the simple field contained
+    in $K$.
+    \begin{enumerate}
+      \item $\acl_K\colon \cP(\cU)\to\cP(\cU)$ is the \emph{algebraic closure
+        operator} over $K$. For a subset $A\subseteq U$ (not necessarily a subfield!) 
+        $\acl_K\colon A\mapsto K(A)^{\alg}\subseteq U$.
+      \item $A\subseteq U$ is \emph{algebraically closed over $K$} when $A =
+        \acl_K(A)$.
+    \end{enumerate}
+  \end{definition}
+  
+  In general, the algebraic closure of $A$ over $K$ can be defined as the set of elements that
+  have finite type over $K\cup A$ (whatever that means). In the case of fields
+  it's equivalent to being algebraic (hence the name).
+
+  \begin{remark}[Properties of $\acl_K$]
+    \label{remark:acl-props}
+    \mbox{}
+    \begin{enumerate}
+      \item $\acl_K(\emptyset) = K$,
+      \item $A\subseteq \acl_K(A)$,
+      \item $A\subseteq B$ $\Rightarrow$ $\acl_K(A)\subseteq\acl_K(B)$, i.e.
+        $\acl_K$ is monotonic,
+      \item $\acl_K(\acl_K(A)) = \acl_K(A)$, i.e. $\acl_K$ is idempotent,
+      \item $\displaystyle \acl_K(A) = \bigcup_{\underset{\text{finite}}{A_0\subset A}}
+        \acl_K(A_0)$
+      \item (\emph{exchange property}) if
+        $a\in\acl_K(A\cup\{b\})\setminus\acl_K(A)$, then
+        $b\in\acl_K(A\cup\{a\})$.
+    \end{enumerate}
+  \end{remark}
+
+  \begin{definition}
+    \mbox{}
+    \begin{enumerate}
+      \item $A\subset \cU$ is \emph{algebraically independent} over $K$ when
+        $\forall a\in A$ $a\notin \acl_K(A\setminus\{a\})$. This is equivalent
+        to saying that $\forall n\forall \underset{\text{pairwise
+        distinct}}{a_1,\ldots,a_n}\in A$ $\forall W(X_1,\ldots, X_n)\in
+        K[\bar{X}]$ $W(\bar{a}) = 0\Leftrightarrow W = 0$.
+      \item $A$ is a \emph{transcendence basis} of $B\subset U$ over $K$, when
+        $A$ is algebraically independent over $K$ and $A\subseteq B\subseteq
+        \acl_K(A)$.
+      \item $\trdeg_K(B) = $ cardinality of any basis of $B/K$, is the
+        \emph{transcendence degree} of $B$ over $K$.
+    \end{enumerate}
+  \end{definition}
+
+  Of course, the third element is the most controversial. First of all,
+  existance of a transcendence basis is not obvious, not to mention that all
+  transcendence bases have the same cardinality. The next theorem clarifies
+  that.
+
+  \begin{theorem}
+    Let $A\subseteq B\subseteq \cU$ and $A$ algebraically independent over $K$.
+    Then there is $A'$ such that $A\subseteq A'\subseteq B$ such that it is a
+    transcendence basis over $B$. Moreover, all transcendence bases of $B$ over
+    $K$ are equinumerous.
+  \end{theorem}
+
+  The proof of this theorem is quite similar to the proof of the same fact in
+  linear spaces.
+
+  \begin{example}
+    \mbox{}
+    \begin{enumerate}
+      \item Let $K$ be a field, $X_i, i\in I$ be set of variables and $M=K(X_i, i\in
+        I)^{\alg}$. Then $\{X_i,i\in I\}\subseteq M$ is algebraically independent
+        over $K$ and $\trdeg_K(M) = \left|I\right|$.
+      \item Let $K\subseteq L\subseteq \cU$ and $\{a_i,i\in I\}$ be a
+        transcendence basis of $L$ over $K$. Then $K(a_i,i\in I)\cong_K
+        K(X_i,i\in I)$ and $K\underset{\text{purely trans.}}{\subseteq} K(a_i,i\in
+        I)\underset{\text{alg.}}{\subseteq} L$
+    \end{enumerate}
+  \end{example}
+
+  \begin{example}
+    Let us find $\trdeg_\bQ(\bC)$ and the size of $\Aut(\bC)$.
+    \begin{enumerate}
+      \item 
+        Let $B\subset \bC$ be transcendence basis of
+        $\bC$ over $\bQ$. Then 
+        \[
+          \bC = \bC^{\alg} \\
+               = \acl_\bQ(B) \\
+              \overset{\ref{remark:acl-props}}{=}
+          \bigcup_{\underset{\text{finite}}{B_0\subseteq B}} \acl_\bQ(B_0),
+        \]
+        hence
+        \[
+          2^{\aleph_0} = \left|\bC\right| =
+          \left|\bigcup_{\underset{\text{finite}}{B_0\subseteq B}}
+          \acl_\bQ(B_0)\right|
+          \le \sum_{\underset{\text{finite}}{B_0\subseteq B}}
+          \left|\acl_\bQ(B_0)\right| = 2^{\aleph_0}.
+        \]
+        The last equality comes from the fact that $\acl_\bQ(B_0)$ is countable
+        comes from the fact that $|\bQ(B_0)| = |\bQ|$ and that the size of the
+        algebraic closure is the size of the field. So, there must be $2^{\aleph_0}$
+        finite subsets of $B$, hence $|B| = 2^{\aleph_0}$.
+      \item 
+        From the previous point we know that $\bC = \bQ(b, b\in B)$. But $\bQ(b,
+        b\in B)\cong\bQ(X_b, b\in B)$, so any permutation of $B$ gives another
+        automorphisms of $\bC$. Hence 
+        \[
+          2^{2^{\aleph_0}} = \left|\bC^\bC\right| \ge \left|\Aut(\bC)\right| \ge
+          \left|\Sym(B)\right| = 2^{2^{\aleph_0}}.
+        \]
+    \end{enumerate}
+  \end{example}
 \end{document}
diff --git a/semestr2/algebra2r/notatki/sections/norm-and-trace.tex b/semestr2/algebra2r/notatki/sections/norm-and-trace.tex
index dd1c145..e75be20 100644
--- a/semestr2/algebra2r/notatki/sections/norm-and-trace.tex
+++ b/semestr2/algebra2r/notatki/sections/norm-and-trace.tex
@@ -1,4 +1,4 @@
-\documentclass[../notes.tex]{subfiles}
+\documentclass[../notatki.tex]{subfiles}
 \begin{document}
   Let $V$ be a linear space over $K$ with $\dim V < \infty$, let $f\colon V\to
   V$ be linear map, $B\subseteq V$ basis. We denote the determinant of $f$ as
diff --git a/semestr2/algebra2r/notatki/sections/prelim.tex b/semestr2/algebra2r/notatki/sections/prelim.tex
index d3e4b3c..c16bb2c 100644
--- a/semestr2/algebra2r/notatki/sections/prelim.tex
+++ b/semestr2/algebra2r/notatki/sections/prelim.tex
@@ -1,4 +1,4 @@
-\documentclass[../notes.tex]{subfiles}
+\documentclass[../notatki.tex]{subfiles}
 \begin{document}
   \begin{definition}
     Field $K$ is \emph{simple} if it has no proper subfield. 
diff --git a/semestr2/algebra2r/notatki/sections/unity-roots.tex b/semestr2/algebra2r/notatki/sections/unity-roots.tex
index e3248c3..b1fde80 100644
--- a/semestr2/algebra2r/notatki/sections/unity-roots.tex
+++ b/semestr2/algebra2r/notatki/sections/unity-roots.tex
@@ -1,4 +1,4 @@
-\documentclass[../notes.tex]{subfiles}
+\documentclass[../notatki.tex]{subfiles}
 \begin{document}
   \begin{definition}
     Let $R$ be a commutative ring with unity.