notes cd

6 files changed, 50 insertions, 8 deletions

diff --git a/semestr2/algebra2r/notatki/notatki.pdf b/semestr2/algebra2r/notatki/notatki.pdf
index 948deb6..da30018 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 18c38c9..276cb6e 100644
--- a/semestr2/algebra2r/notatki/notatki.tex
+++ b/semestr2/algebra2r/notatki/notatki.tex
@@ -51,6 +51,7 @@
 \DeclareMathOperator{\Stab}{Stab} 
 \DeclareMathOperator{\st}{st}
 \DeclareMathOperator{\Flim}{Flim} 
+\DeclareMathOperator{\Gal}{Gal} 
 \DeclareMathOperator{\Int}{{Int}}
 \DeclareMathOperator{\ord}{{ord}}
 \DeclareMathOperator{\Tr}{{Tr}}
@@ -88,6 +89,8 @@
 
 
 \DeclareMathOperator{\im}{{Im}} 
+\DeclareMathOperator{\Fix}{{Fix}} 
+\DeclareMathOperator{\Sym}{{Sym}} 
 \DeclareMathOperator{\sep}{{sep}} 
 \DeclareMathOperator{\rad}{{rad}} 
 \DeclareMathOperator{\Char}{{char}} 
@@ -154,7 +157,7 @@
   \vspace{3cm}
   \vfill
   \begin{center}
-	{\large Wrocław 2022}\\
+	{\large Wrocław 2023}\\
   \end{center}
 
   \newpage
@@ -164,12 +167,18 @@
   I assume you're familiar with the notion of ring, polynomial ring, field,
   field extension, homomorphism, etc. 
 
-  The following holds:
   Also, it was very hard to order the topics, as they interleave a lot, so there
   may be places where a proof of statement of a theorem refers to some notion
   from the further parts of the note. However, I assume that reader has some
   knowledge of Galois theory and uses the note as a reminder for the exam.
 
+  Everything written in red is subject to change. If you have idea how to fill
+  the gaps, then don't hesitate to let me now.
+
+  Checkout my page: \url{https://framal.xyz}, 
+  \href{franciszek.malinka@gmail.com}{write me an email} or 
+  \href{https://www.buymeacoffee.com/framal}{buy me a coffee}.
+
   \section{Preliminaries} 
   \label{section:preliminaries}
   \subfile{sections/prelim}
@@ -195,8 +204,11 @@
   \label{section:norm-and-trace}
   \subfile{sections/norm-and-trace}
 
+  \section{Galois correspondence}
+  \label{section:galois-cont}
+  \subfile{sections/galois-cont}
+
   \section{Transcendental extensions}
   \label{section:lioville}
   \subfile{sections/lioville}
-
 \end{document}
diff --git a/semestr2/algebra2r/notatki/sections/alg-ext.tex b/semestr2/algebra2r/notatki/sections/alg-ext.tex
index d5e8a93..4e9ae10 100644
--- a/semestr2/algebra2r/notatki/sections/alg-ext.tex
+++ b/semestr2/algebra2r/notatki/sections/alg-ext.tex
@@ -115,6 +115,11 @@
     a_n)$, so $[K(m):K] \le [K'(m):K] = [K'(m):K']\cdot[K':K] < \infty$, as both
     $[K'(m):K'], [K':K]$ are finite.
   \end{proof}
+
+  \begin{example}
+    If $[L\colon K] = p$ for $p$ prime then there are no intermediate fields between $L$ and
+    $K$.
+  \end{example}
   
   \subsection{Algebraic closure}
   \begin{definition}
diff --git a/semestr2/algebra2r/notatki/sections/galois.tex b/semestr2/algebra2r/notatki/sections/galois.tex
index 4931702..5adefb2 100644
--- a/semestr2/algebra2r/notatki/sections/galois.tex
+++ b/semestr2/algebra2r/notatki/sections/galois.tex
@@ -37,7 +37,7 @@
 
   \begin{theorem}
     $K\subset L$ is a normal extension $\Leftrightarrow$  $\forall f\in
-    G(\hat{K}/K)$ $f[L] = L$ $\Leftrightarrow$ $\forall b\in L \, W_b(X)\in
+    G(\hat{K}/K)$ $f[L] = L$ $\Leftrightarrow$ $\forall b\in L \; W_b(X)\in
     K[X]$ splits in $L[X]$ into linear factors (where $W_b(X)$ is the minimal
     polynomial of $b$ over $K$).
   \end{theorem}
@@ -148,7 +148,7 @@
   \begin{proof}{(key ideas)}
     This is very easy in the finite fields. For infinite it's quite tricky and
     technical. The idea is to show that it is true for two separable elements
-    $a,b$, the rest follows through indunction. It turns out that we can construct
+    $a,b$, the rest follows through induction. It turns out that we can construct
     $a^*$ with suitable $c\in K$, such that $a^* = a + cb$. This is great news,
     because it basically means that in general $a^*$ is some linear combination
     of $a_1,\ldots, a_n$. Obviously $K(a^*)\subseteq K(a,b)$ no matter what $c$
diff --git a/semestr2/algebra2r/notatki/sections/norm-and-trace.tex b/semestr2/algebra2r/notatki/sections/norm-and-trace.tex
index 6b06308..dd1c145 100644
--- a/semestr2/algebra2r/notatki/sections/norm-and-trace.tex
+++ b/semestr2/algebra2r/notatki/sections/norm-and-trace.tex
@@ -22,6 +22,28 @@
   \end{definition}
 
   \begin{fact}
+    Assume $a\in L$ algebraic over $K$, $L=K[a]$, let $W(X) = X^n +
+    a_{n-1}X^{n-1}+\ldots+a_1X +a_0$ be the minimal polynomial of $a$ over $K$.
+    Then:
+    \begin{enumerate}
+      \item $\Tr_{L/K}(a) = -a_n$,
+      \item $N_{L/K}(a) = (-1)^n a_0$,
+      \item $W(X) = (-1)^n\phi(x)$, where $\phi(x)$ is the characteristic
+        polynomial of the linear map $f_a$.
+    \end{enumerate}
+  \end{fact}
+
+  \begin{fact}
+    Let $K\subset L\subset M$, $[M\colon K] < \infty$ and $a\in L$. Then:
+    \begin{enumerate}
+      \item $\Tr_{M/K}(a) = [M\colon L]\cdot \Tr_{L/K}(a)$,
+      \item $N_{M/K}(a) = N_{L/K}(a)^{[M\colon L]}$,
+      \item $\Tr_{M/K} = \Tr_{L/K}\circ\Tr_{M/L}$,
+      \item $N_{M/K} = N_{L/K}\circ N_{M/L}$.
+    \end{enumerate}
+  \end{fact}
+
+  \begin{fact}
     Let $\{f_1\ldots, f_k\} = \{f\colon L\to \hat{K}\mid \restr{f}{K} =
     \id_K\}$, $k = \sepdeg{L}{K}$ and $a\in L$. Then:
     \begin{enumerate}
@@ -30,5 +52,8 @@
         the norm).
     \end{enumerate}    
   \end{fact}
-  
+
+  \begin{example}
+    \color{red}{There will be a nice example one day...}
+  \end{example}
 \end{document}
diff --git a/semestr2/algebra2r/notatki/sections/prelim.tex b/semestr2/algebra2r/notatki/sections/prelim.tex
index f3596e3..d3e4b3c 100644
--- a/semestr2/algebra2r/notatki/sections/prelim.tex
+++ b/semestr2/algebra2r/notatki/sections/prelim.tex
@@ -63,7 +63,7 @@
   \begin{definition}
     Let $K\subset L_1$, $K\subset L_2$ field extensions, then:   
     \begin{displaymath}
-      L_1\cong_K l_2 \Leftrightarrow \exists f\colon
+      L_1\cong_K L_2 \Leftrightarrow \exists f\colon
       L_1\xrightarrow{\cong} L_2 \text{ such that } f_{\upharpoonright K} =
       \id_K
     \end{displaymath}
@@ -109,7 +109,7 @@
 
   \begin{proof}
     As $\Char(K) = 0$, all roots of $f$ in $L$ are pairwise disjoint.
-    Let $a_1\,ldots, a_n$ be roots of $f$ in $L$. It's easy to see that
+    Let $a_1,\ldots, a_n$ be roots of $f$ in $L$. It's easy to see that
     $I(a_i/K) = I(a_j / K)$ for any $1\le i, j\le n$. Then by \ref{remark:iso}
     we know that there is an automorphism $f\colon L\to L$ such that $f(a_1) =
     f(a_i)$ for any $1\le i\le n$.