port notes on A-algebras & Noether normalization & integral closures (#3)

* port notes on A-algebras & integral elems

* add R.4.3's proof & 4.4

* add Noether normalization proof

galois-theory-notes.tex

@@ -517,12 +517,12 @@
       Note that $HN = \{ hn : h\in H, n\in N \}$. Let $h_1 n_1, h_2 n_2 \in HN$.
 
       Since $N$ normal $\Longrightarrow~ h_2^{-1} n_1 h_2 \in N$, so
-      $$(h_1 n_1)(h_2 n_2) = h_1 h_2 (h_2^{-1} n_1 h_2) \in HN$$
+      $$(h_1 n_1)(h_2 n_2) = h_1 h_2 (h_2^{-1} n_1 h_2) \cdot n_2 \in HN$$
 
       [Recall: since $N \triangleleft G$, $gN=Ng ~\forall g \in G$ $\Longrightarrow gn=n'g$ for some $n' \in N$.]
 
       To see that $(hn)^{-1} \in HN$:\\
-      since $(hn)^{-1} = n^{-1} h^{-1} = h^{-1} (h n^{-1} h^{-1})$, thus $(hn)^{-1} \in HN$.
+      since $(hn)^{-1} = h^{-1} n^{-1} = h^{-1} (h n^{-1} h^{-1})$, thus $(hn)^{-1} \in HN$.
 
       Thus $HN \subseteq G$.