07-401/Class Notes for April 11

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In Preparation

The information below is preliminary and cannot be trusted! (v)

The goal of today's class is to prove a (weak but strong enough) form of the Fundamental Theorem of Galois Theory as follows:

Theorem. Let [math]\displaystyle{ F }[/math] be a field of characteristic 0 and let [math]\displaystyle{ E }[/math] be a splitting field over [math]\displaystyle{ F }[/math]. Then there is a correspondence between the set [math]\displaystyle{ \{K:E/K/F\} }[/math] of intermediate field extensions [math]\displaystyle{ K }[/math] lying between [math]\displaystyle{ F }[/math] and [math]\displaystyle{ E }[/math] and the set [math]\displaystyle{ \{H:H\lt \operatorname{Gal}(E/F)\} }[/math] of subgroups [math]\displaystyle{ H }[/math] of the Galois group [math]\displaystyle{ \operatorname{Gal}(E/F) }[/math] of the original extension [math]\displaystyle{ E/F }[/math]:

[math]\displaystyle{ \{K:E/K/F\}\quad\leftrightarrow\quad\{H:H\lt \operatorname{Gal}(E/F)\} }[/math].

The bijection is given by mapping every intermediate extension [math]\displaystyle{ K }[/math] to the subgroup [math]\displaystyle{ \operatorname{Gal}(E/K) }[/math] of elements in [math]\displaystyle{ \operatorname{Gal}(E/F) }[/math] that preserve [math]\displaystyle{ K }[/math],

[math]\displaystyle{ K\mapsto\operatorname{Gal}(E/K) }[/math],

and reversely, by mapping every subgroup [math]\displaystyle{ H }[/math] of [math]\displaystyle{ \operatorname{Gal}(E/F) }[/math] to its fixed field [math]\displaystyle{ E_H }[/math]:

[math]\displaystyle{ H\mapsto E_H }[/math].

Furthermore, this correspondence has the following further properties:

1. It is inclusion-reversing: if [math]\displaystyle{ H_1\subset H_2 }[/math] then [math]\displaystyle{ E_{H_1}\supset E_{H_2} }[/math] and if [math]\displaystyle{ K_1\subset K_2 }[/math] then [math]\displaystyle{ \operatorname{Gal}(E/K_1)\gt \operatorname{Gal}(E/K_1) }[/math].
2. It is degree/index respecting: [math]\displaystyle{ [E:K]=|\operatorname{Gal}(E/K)| }[/math] and [math]\displaystyle{ [K:F]=[\operatorname{Gal}(E/F):\operatorname{Gal}(E/K)] }[/math].
3. Splitting fields correspond to normal subgroups: If [math]\displaystyle{ K }[/math] in [math]\displaystyle{ E/K/F }[/math] is a splitting field then [math]\displaystyle{ \operatorname{Gal}(E/K) }[/math] is normal in [math]\displaystyle{ \operatorname{Gal}(E/F) }[/math] and [math]\displaystyle{ \operatorname{Gal}(K/F)\cong\operatorname{Gal}(E/F)/\operatorname{Gal}(E/K) }[/math].