port initial notes on commutative algebra (ideals, modules, Noetherian rings) (#1)

* port initial notes on commutative algebra: ideals & modules, Nakayama's lemma, etc

* port notes on Noetherian rings&modules

* add ideals related definitions

* improve Cayley-Hamilton proof (specially determinant trick explanation)

* polishing

* add typos detection

* add some exercises, and proof of Z and K[X} being PID

galois-theory-notes.tex

@@ -277,7 +277,7 @@
 \end{thm}
 
 \begin{thm}{9.10}
-  An irreducible polynomial $f \in K[t]$ ($K \subseteq \mathbb{C}$) is \emph{separable over} $K$ if it has simple zeros in $\mathbb{C}$, or equivelently, simple zeros in its splitting field.
+  An irreducible polynomial $f \in K[t]$ ($K \subseteq \mathbb{C}$) is \emph{separable over} $K$ if it has simple zeros in $\mathbb{C}$, or equivalently, simple zeros in its splitting field.
 \end{thm}
 
 \begin{lemma}{9.13}
@@ -809,7 +809,7 @@
 
   Let $f_i$ be the minimal polynimal of $\alpha_i$ over $K$.
 
-  Then, $M \supseteq L$ is spliting field of $\Prod_{i=1}^r f_i$, since $M$ is normal enclosure of $L:K$.
+  Then, $M \supseteq L$ is splitting field of $\Prod_{i=1}^r f_i$, since $M$ is normal enclosure of $L:K$.
 
   For every zero $\beta_{ij}$ of $f_i$ in $M$,\\
   $\exists$ an isomorphism $\sigma: K(\alpha_i) \longrightarrow K(\beta_{ij})$ by Corollary \ref{5.13}.
@@ -1065,7 +1065,7 @@ For $n \geq 3, ~~\mathbb{D}_n \subseteq \mathbb{S}_n$ (subgroup of the Symmetry
   then, $\exists$ at least one $c$ in $(a,b)$ such that $Df(c)=0$.
 \end{thm}
 \begin{proof} (proof source: cue math website)
-  Notice that when $Df(x_i)=0$ occours, is a maximum or minimum (extrema) value of $f$.
+  Notice that when $Df(x_i)=0$ occurs, is a maximum or minimum (extrema) value of $f$.
   $\Longrightarrow$ if a function is continuous, it is guaranteed to have both a maximum and a minimum point in the interval.\\
 
   Two possibilities:
@@ -1076,9 +1076,9 @@ For $n \geq 3, ~~\mathbb{D}_n \subseteq \mathbb{S}_n$ (subgroup of the Symmetry
     since $f$ not constant, must change directions in ordder to start and end at the same $y$-value ($f(a)=f(b)$).\\
     Thus at some point between $a$ and $b$ it will either have a minimum, maximum or both.
     \begin{enumerate}[a.]
-      \item does the maximum occour at a point where $Df > 0$?\\
+      \item does the maximum occur at a point where $Df > 0$?\\
 	No, because if $Df > 0$, then $f$ is increasing, but it can not increase since we're at its maximum point.
-      \item does the maximum occour at a point where $Df < 0$?\\
+      \item does the maximum occur at a point where $Df < 0$?\\
 	No, because if $Df < 0$, then $f$ is deccreasing, which means that at our left it was larger, but we're at a maximum point, so a contradiction.
     \end{enumerate}
     Same with minimus.\\