SQUARE45

Let

L/K

be a finite, separable, and normal field extension. The Galois group of this extension is defined as

\text{Gal}(L/K) = \text{Aut}_K(L)

, the group of all

K

-automorphisms of

L

. The fundamental theorem of Galois Theory establishes a canonical, inclusion-reversing bijection (an anti-isomorphism) between the set of subgroups of

\text{Gal}(L/K)

and the set of intermediate fields

M

such that

K \subseteq M \subseteq L

. Specifically, for any subgroup

H \subseteq \text{Gal}(L/K)

, the corresponding intermediate field is the fixed field

L^H = \{x \in L \mid \sigma(x) = x \text{ for all } \sigma \in H\}

. Conversely, for any intermediate field

M

, the corresponding subgroup is

\text{Gal}(L/M) = \text{Aut}_M(L)

. The correspondence is given by the pair of inverse maps:

H \mapsto L^H

and

M \mapsto \text{Gal}(L/M)

. Furthermore, the degree of the extension

[L:M]

equals the order of the corresponding group

|\text{Gal}(L/M)|

, and the degree

[M:K]

equals the index

[\text{Gal}(L/K):\text{Gal}(L/M)]

. This establishes the isomorphism:

\text{Gal}(L/K) / \text{Gal}(L/M) \cong \text{Gal}(M/K)

Galois Theory

Sequence of Expressions

Galois Theory (Definition)

Fundamental Theorem of Galois Theory