5 files changed, 19 insertions, 12 deletions
diff --git a/semestr2/algebra2r/notatki/LICENSE b/semestr2/algebra2r/notatki/LICENSE
new file mode 100644
index 0000000..070f4e7
--- /dev/null
+++ b/semestr2/algebra2r/notatki/LICENSE
@@ -0,0 +1,7 @@
+Copyright 2023 Franciszek Malinka 
+
+Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the “Software”), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:
+
+The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.
+
+THE SOFTWARE IS PROVIDED “AS IS”, WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.
diff --git a/semestr2/algebra2r/notatki/notatki.pdf b/semestr2/algebra2r/notatki/notatki.pdf
index a8dba76..186cd0c 100644
--- a/semestr2/algebra2r/notatki/notatki.pdf
+++ b/semestr2/algebra2r/notatki/notatki.pdf
diff --git a/semestr2/algebra2r/notatki/sections/galois-cont.tex b/semestr2/algebra2r/notatki/sections/galois-cont.tex
index 5a154d3..fe1ee3e 100644
--- a/semestr2/algebra2r/notatki/sections/galois-cont.tex
+++ b/semestr2/algebra2r/notatki/sections/galois-cont.tex
@@ -14,7 +14,7 @@
   \end{definition}
   \begin{definition}
     Let $G<\Aut(L)$. Then $L^G\defeq \{a\in L\mid \forall f\in G$ $f(a) = a\} =
-    \bigcup_{f\in G}\Fix(f)$ is the \emph{fixed-point field of $G$}.
+    \bigcap_{f\in G}\Fix(f)$ is the \emph{fixed-point field of $G$}.
   \end{definition}
 
   We get a nice equivalent definition for Galois extensions with the notion of
@@ -73,7 +73,7 @@
     $W_a(X)$ that reside in $L$. Look at $V(X) =
     (X-a_1)\cdot\ldots\cdot(X-a_k)\in L[X]$, of course $V(X)\mid W_a(X)$ in
     $L[X]$, but as any $f\in G(L/K)$ permutes $\{a_1,\ldots,a_k\}$ we get that
-    $f(V) = V$, hence $V(X)\in L^{G(L/K)[X} = K[X]$ as $K\subset L$ is Galois.
+    $f(V) = V$, hence $V(X)\in L^{G(L/K)}[X] = K[X]$ as $K\subset L$ is Galois.
     But $W_a$ is minimal, hence $V = W_a$, so all roots of $W_a$ are in $L$ and
     every root is simple, so $L$ is normal and separable over $K$.
 
@@ -102,7 +102,7 @@
 
   The following theorem is also extremely important for studying examples of
   Galois extensions. It is a corollary from Abel's primitive element
-  theorem\ref{theorem:abel}.
+  theorem \ref{theorem:abel}.
   \begin{theorem}[Artin's]
     \label{theorem:artins}
     Let $G<\Aut(L)$ be a finite subgroup. Then $L^G\subset L$ is Galois
@@ -163,7 +163,7 @@
     The other way, let $H\in\cG$, then
     $H\overset{\Lambda}{\longmapsto}L^H\overset{\Gamma}{\longmapsto}G(L/L^H) =
     H$, where the last equality follows from the Artin's theorem
-    \ref{theorem:artins}. Hence $\gamma\circ\Lambda = \id_\cG$.
+    \ref{theorem:artins}. Hence $\Gamma\circ\Lambda = \id_\cG$.
   \end{proof}
 
   We will look on example usages of the theorem a little bit later. Now we'll
diff --git a/semestr2/algebra2r/notatki/sections/galois.tex b/semestr2/algebra2r/notatki/sections/galois.tex
index e15aa10..d06a0fa 100644
--- a/semestr2/algebra2r/notatki/sections/galois.tex
+++ b/semestr2/algebra2r/notatki/sections/galois.tex
@@ -71,7 +71,7 @@
 
   \begin{definition}
     The field $L_1\overset{\text{def}}{=} \text{ field generated by
-    }\bigcup\{f[L]\colon f\in G(\hat{K}/K)\}$ is the \emph{normal closure} of
+    }\bigcap\{f[L]\colon f\in G(\hat{K}/K)\}$ is the \emph{normal closure} of
     $L$ in $\hat{K}$.
   \end{definition}
 
@@ -101,18 +101,18 @@
    \begin{enumerate}
      \item $W(X)$ is separable $\Leftrightarrow$ $W(X)$ and $W'(X)$ (the formal
        derivative) are coprime in $K[X]$,
-     \item Every $W(X)$ is separable when $\Char K = 0$
-      \item When $\Char K = p >0$, then $W$ is not separable if and only if
+     \item Every irreducible $W(X)$ is separable when $\Char K = 0$
+      \item When $\Char K = p >0$, then $W$ is inseparable if and only if
         $W(X) = V(X^p)$ for some $V\in K[X]$.
    \end{enumerate} 
   \end{remark}
 
-  When $K\subset L_1\subset L$ and $K\subset L$ then obviously $L_1\subset L$ is
+  When $K\subset L_1\subset L$ and $K\subset L$ separable then obviously $L_1\subset L$ is
   also separable. Also, in $\Char = 0$ all algebraic extensions are separable.
   Extensions of finite fields are also separable, as the bigger field is the
   splitting field of some polynomial $X^{p^n}-X$, which has $p^n$ simple roots.
 
-  \begin{example}[of not separable extensions]
+  \begin{example}[of inseparable extensions]
     Let $K = F_p(X)$, i.e. the field of rational functions. Let $L =
     K(\sqrt[p]{X})$. Then $W_{\sqrt[p]{X}}(T) = T^p - X\in K[T]$ is
     irreducible ,but $W_{\sqrt[p]{X}}(T) = (T-\sqrt[p]{X})^p$.
@@ -233,10 +233,10 @@
     \begin{enumerate}
       \item $\sepdeg{L}{K} \defeq [\sep_L(K):K]$ is the \emph{separable degree} 
         of $L$ over $K$,
-      \item $\raddeg{L}{K} \defeq [L\colon\rad_L(K)]$ is the \emph{radical degree}
+      \item $\raddeg{L}{K} \defeq [L\colon\sep_L(K)]$ is the \emph{radical degree}
         of $L$ over $K$.
     \end{enumerate}
-     Note $\raddeg{L}{K}\cdot\sep{L}{K} = [L\colon K]$.
+     Note $\raddeg{L}{K}\cdot\sepdeg{L}{K} = [L\colon K]$.
   \end{definition}
 
   \begin{remark}
diff --git a/semestr2/algebra2r/notatki/sections/modules.tex b/semestr2/algebra2r/notatki/sections/modules.tex
index a8b2348..385a56c 100644
--- a/semestr2/algebra2r/notatki/sections/modules.tex
+++ b/semestr2/algebra2r/notatki/sections/modules.tex
@@ -384,7 +384,7 @@
   $\bZ$-module.
 
   \begin{corollary}
-    Let $R$ be a Noetherian ring and $\{Q_j\}_\{j\in J\}$ be a family of
+    Let $R$ be a Noetherian ring and $\{Q_j\}_{j\in J}$ be a family of
     injective $R$-modules. Then the direct sum $\bigoplus_j Q_j$ is also
     injective.
   \end{corollary}