The Fundamental Theorem of Galois Theory

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This post is being revised. The text below is the version as of 2026-06-05, so some statements may be out of date and references to this post from other posts may not line up.

This post was machine-translated from the Korean original by Marvin (via Kimi). It may contain errors or awkward phrasing — 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