Properties of The Correspondence
The correspondence has the following useful properties.
- It is inclusion-reversing. The inclusion of subgroups H1 ⊆ H2 holds if and only if the inclusion of fields EH1 ⊇ EH2 holds.
- Degrees of extensions are related to orders of groups, in a manner consistent with the inclusion-reversing property. Specifically, if H is a subgroup of Gal(E /F ), then |H| = and |Gal(E /F )/H| = .
- The field EH is a normal extension of F (or, equivalently, Galois extension, since any subextension of a separable extension is separable) if and only if H is a normal subgroup of Gal(E /F ). In this case, the restriction of the elements of Gal(E /F ) to EH induces an isomorphism between Gal(EH/F ) and the quotient group Gal(E /F )/H.
Read more about this topic: Fundamental Theorem Of Galois Theory
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