The Fundamental Theorem of Galois Theory

This post was translated from Korean by LLM (Kimi). The translation may contain errors or awkward sentences. The Korean original is the source of truth.

We can now finally prove the fundamental theorem of Galois theory.

Theorem 1 Consider a Galois extension \(\mathbb{L}/\mathbb{K}\) of a field \(\mathbb{K}\) and its Galois group \(\Gamma=\Gal(\mathbb{L}/\mathbb{K})\). Let \(\mathscr{K}\) be the collection of subextensions of \(\mathbb{L}\), and let \(\mathscr{G}\) be the collection of closed subgroups of \(\Gamma\). Then the two functions between \(\mathscr{K}\) and \(\mathscr{G}\)

\[k:\mathscr{G}\rightarrow\mathscr{K};\qquad G\mapsto k(G)\text{ the field of invariants of $G$}\]

and

\[g:\mathscr{K}\rightarrow\mathscr{G};\qquad \mathbb{M}\mapsto g(\mathbb{M})\text{ the group of $\mathbb{M}$-automorphisms of $L$}\]

are inverses of each other.

To prove this, we divide the proof into two steps as follows.

Lemma 2 For any subextension \(\mathbb{M}\in \mathscr{K}\), \(\mathbb{L}/\mathbb{M}\) is also a Galois extension. In this case, if we regard the Galois group \(\Gal(\mathbb{L}/\mathbb{M})\) as a subgroup of \(\Gal(\mathbb{L}/\mathbb{K})\) in the obvious way, it is a closed subgroup of \(\Gal(\mathbb{L}/\mathbb{K})\), and therefore \(g\) is well-defined.

Proof