update

3 files changed, 31 insertions, 7 deletions

diff --git a/semestr2/algebra2r/notatki/notatki.pdf b/semestr2/algebra2r/notatki/notatki.pdf
index 15ecfe1..d772f11 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 f715517..f94b37d 100644
--- a/semestr2/algebra2r/notatki/notatki.tex
+++ b/semestr2/algebra2r/notatki/notatki.tex
@@ -199,6 +199,7 @@
   \href{https://www.buymeacoffee.com/framal}{buy me a coffee}.
 
   \tableofcontents
+  \newpage
 
   \section{Preliminaries} 
   \label{section:preliminaries}
diff --git a/semestr2/algebra2r/notatki/sections/galois-cont.tex b/semestr2/algebra2r/notatki/sections/galois-cont.tex
index 2896ce9..5160cdf 100644
--- a/semestr2/algebra2r/notatki/sections/galois-cont.tex
+++ b/semestr2/algebra2r/notatki/sections/galois-cont.tex
@@ -274,8 +274,26 @@
     Let's find all intermediate fields of $\bQ\subset \bQ(\sqrt[3]{2},
     \zeta)$. We already know that $\Gal(\bQ(\sqrt[3]{2},\zeta)/\bQ)\cong D_3$,
     i.e. the dihedral group of triangle. It has four subgroups, namely $\{\id, r,
-    r^2\}$, $\{\id, s\}$, $\{\id, rs\}$ and $\{\id, r^2s\}$. So, we have four 
-    intermediate fields corresponding two these subgroups. 
+    r^2\}$, $\{\id, s\}$, $\{\id, rs\}$ and $\{\id, r^2s\}$. See that the
+    smallest subgroup is on top, as it corresponds to the largest field, i.e. $\bQ(\sqrt[3]{2},
+    \zeta)$
+
+    \begin{center}
+        \begin{tikzcd}
+          & &  \{\id\} & \\
+          \langle s \rangle \arrow[dash, urr] & \langle rs \rangle \arrow[ur,
+          dash] & \langle r^s \rangle \arrow[dash, u] & \\
+          & & & \langle r \rangle \arrow[dash, uul] \\
+          & & \langle s, r\rangle \arrow[uull, dash, "3"] \arrow[uul, dash, "3"]
+          \arrow[uu, dash, "3"] \arrow[ur, dash, "2"'] &
+        \end{tikzcd}
+    \end{center}
+
+    So, we have four 
+    intermediate fields corresponding two these subgroups. Orders of these
+    groups are somewhat opposite to the degrees of the extensions that
+    correspond to them, i.e. there should be three extensions of degree $3$ and
+    one extension of degree $2$. 
 
     Now we need to find the intermediate fields that correspond to the subgroups
     of the Galois group. In this case, I like to think of roots of the $X^3-2$
@@ -300,14 +318,19 @@
     \begin{center}
         \begin{tikzcd}
           & &  \bQ(\sqrt[3]{2}, \zeta) & \\
-          & &  & \bQ(\sqrt[3]{2}) \arrow[ul, dash] \\
-          \bQ(\zeta) \arrow[uurr, dash] & \bQ(\zeta\sqrt[3]{2}) \arrow[uur,
+          \bQ(\sqrt[3]{2}) \arrow[urr, dash] & \bQ(\zeta\sqrt[3]{2}) \arrow[ur,
           dash] &
-          \bQ(\zeta^2\sqrt[3]{2}) \arrow[uu, dash] & \\
-          & &  \bQ \arrow[ull, dash, "2"]\arrow[u, dash, "2"] \arrow[ul,
-          dash, "2"'] \arrow[uur, dash, "3"'] &\\
+          \bQ(\zeta^2\sqrt[3]{2}) \arrow[u, dash] & \\
+          & &  & \bQ(\zeta) \arrow[uul, dash] \\
+          & &  \bQ \arrow[uull, dash, "3"]\arrow[uu, dash, "3"] \arrow[uul,
+          dash, "3"'] \arrow[ur, dash, "2"'] &\\
         \end{tikzcd}
     \end{center}
+
+    All that's left is to check if we correctly assigned the intermediate fields
+    to the corresponding groups. For example, $\sqrt[3]{2}$ is invariant on the
+    $s$ automorphism, so this one is correct. It's similar to check for the
+    other two that we assigned them correctly.
   \end{example}
 
   \newcommand{\bQQ}{\bQ(\sqrt[3]{2}, \sqrt{3})}