Fundamental Theorem of Galois Theory - Properties of The Correspondence

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.