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].