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单词 ProofOfGaloisGroupOfTheCompositumOfTwoGaloisExtensions
释义

proof of Galois group of the compositum of two Galois extensions


Proof.

Consider the diagram

\\xymatrix@R1pc@C1pc&\\ar@-[ld]\\ar@-[rd]EF\\ar@-[rd]E&&\\ar@-[ld]F&\\ar@-[d]EF&K

(1): Let p(x)K[x] with a root αEF. Then since E (resp. F) is Galois over K, all the roots of p lie in E (resp. F) and thus in EF. The result follows.

(2): We first show that EF is Galois over K. Choose separable polynomialsMathworldPlanetmath p(x),q(x)K[x] so that E (resp. F) is a splitting fieldMathworldPlanetmath for p (resp. q). Then EF is a splitting field for the squarefree part of pq, which is separable since it is squarefreeMathworldPlanetmath and since p(x),q(x) are separable.

Now, define

θ:Gal(EF/K)Gal(E/K)×Gal(F/K):σ(σ|E,σ|F)

This map is a group homomorphismMathworldPlanetmath; its kernel is precisely those elements that leave both E and F fixed. Any such element must thus leave EF fixed, so that θ is injective. The image obviously lies in

H={(σ,τ):σ|EF=τ|EF}

by construction: (σ|E)|EF=σ|EF=(σ|F)|EF. We will show that H is precisely the image of θ by showing that the order of H is the same as the index of the field extension [EF:K].

For each σGal(E/K), there are precisely |Gal(F/EF)| elements of Gal(F/K) whose restrictions to EF are σ|EF. Thus directly from the definition of H,

|H|=|Gal(E/K)||Gal(F/EF)|=|Gal(E/K)||Gal(F/K)||Gal((EF)/K)|

By the corollary to the theorem regarding the compositum of a Galois extensionMathworldPlanetmath and another extension (http://planetmath.org/CorollaryToTheCompositumOfAGaloisExtensionAndAnotherExtensionIsGalois), we have

[EF:K]=[EF:F][F:K]=[E:EF][F:K]=[E:K][F:K][EF:K]

so that

|H|=[EF:K]

References

  • 1 Dummit, D., Foote, R.M., Abstract Algebra, Third Edition, Wiley, 2004.
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更新时间:2025/5/4 12:07:29