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add Noether normalization proof

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arnaucube
2026-01-02 19:32:10 +01:00
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galois-theory-notes.tex
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@@ -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 $HN \subseteq G$.