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Fundamental Theorem of Galois Theory


For a Galois extension field K of a field F, the fundamental theorem of Galois theory states that the subgroups of the Galois group G=Gal(K/F) correspond with the subfields of K containing F. If the subfield L corresponds to the subgroup H, then the extension field degree of K over L is the group order of H,

|K:L|=|H|
(1)
|L:F|=|G:H|.
(2)

Suppose F subset E subset L subset K, then E and L correspond to subgroups H_E and H_L of G such that H_E is a subgroup of H_L. Also, H_E is a normal subgroup iff E is a Galois extension field. Since any subfield of a separable extension, which the Galois extension field K must be, is also separable, E is Galois iff E is a normal extension of F. So normal extensions correspond to normal subgroups. When H_E is normal, then

 Gal(E/F)=G/H
(3)

as the quotient group of the group action of G on K.

FundamentalTheoremofGalois

According to the fundamental theorem, there is a one-one correspondence between subgroups of the Galois group Gal(L/K) and subfields of L containing K. For example, for the number field L shown above, the only automorphisms of L (keeping K=Q fixed) are the identity, sigma, tau, and sigmatau, so these form the Galois group Gal(L/K) (which is generated by sigma and tau). In particular, the generators sigma and tau of G are as follows: sigma maps sqrt(3) to -sqrt(3), sqrt(6) to -sqrt(6), and fixes sqrt(2); tau maps sqrt(2) to -sqrt(2), sqrt(6) to -sqrt(6) and fixes sqrt(3); and sigmatau maps sqrt(2) to -sqrt(2), sqrt(3) to -sqrt(3) and fixes sqrt(6).

For example, consider the Galois extension field

K=Q(2^(1/3),omega)
(4)
={a_1+a_2omega+a_32^(1/3)+a_42^(1/3)omega+a_52^(2/3) +a_62^(2/3)omega:a_i in Q}
(5)

over F=Q, which has extension field degree six. That is, it is a six-dimensional vector space over the rationals.


See also

Field Automorphism, Galois Theory, Simple Extension

Portions of this entry contributed by Todd Rowland

Portions of this entry contributed by David Terr

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Cite this as:

Rowland, Todd; Terr, David; and Weisstein, Eric W. "Fundamental Theorem of Galois Theory." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/FundamentalTheoremofGaloisTheory.html

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