characterization of abelian extensions of exponent n

Theorem 1.

Let K be a field containing the nth roots of unityMathworldPlanetmath, with characteristicPlanetmathPlanetmath not dividing n. Let L be a finite extensionMathworldPlanetmath of K. Then the following are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath:

  1. 1.

    L/K is Galois with abelianMathworldPlanetmath Galois groupMathworldPlanetmath of exponent dividing n

  2. 2.

    L=K(a1n,,akn) for some aiK.


Let ζK be a primitive nth root of unity.

(21): Choose αiL such that αin=aiK. Then for each i, the elements αi,ζαi,,ζn-1αi are distinct and are all the roots of xn-ai in L. Thus xn-ai is separablePlanetmathPlanetmath over K and splits in L, so that L is the splitting fieldMathworldPlanetmath of the set of polynomialsMathworldPlanetmathPlanetmathPlanetmath {xn-ai 1ik}. Thus L/K is Galois. Given σGal(L/K), for each i we have σ(αi)=ζjαi for some 1jn, so that σk(αi)=ζkjαi. It follows that σn is the identityPlanetmathPlanetmathPlanetmathPlanetmath for every σGal(L/K), so that the exponent of Gal(L/K) divides n. It remains to show that Gal(L/K) is abelian; this follows trivially from the simple definition of the Galois action as multiplicationPlanetmathPlanetmath by some nth root of unity: if σ,τGal(L/K) with σ(αi)=ζrαi,τ(αi)=ζsαi, then

(στ)(αi) =σ(ζsαi)=ζsζrαi
(τσ)(αi) =τ(ζrαi)=ζrζsαi

Thus στ=τσ on each αi. But the αi generate L/K, so στ=τσ on L and Gal(L/K) is abelian.

(12): Let G=Gal(L/K), and write G=C1××Cr where each Ci is cyclic; |Ci|=min for each i. For each i, define a subgroupMathworldPlanetmathPlanetmath HiG by


Then G/HiCi. Let Li be the fixed field of Hi. Li is normal over K since Hi is normal in G, and Gal(Li/K)G/HiCi and thus Li/K is cyclic Galois of order mi. K contains the primitive mith root of unity ζn/mi and thus Li=K(αi) for some αiL with αimiK (by Kummer theory). But then also αinK. Then


since any element of the left-hand group fixes each αi and thus fixes Li so is the identity in G/Hi. Thus L=K(α1,,αr). ∎

Corollary 2.

If L/K is the maximal abelian extensionMathworldPlanetmathPlanetmath of K of exponent n, where n is prime to the characteristic of K, then L=K({an}) for some set of aK.


Clearly K({anaK*) is an infiniteMathworldPlanetmathPlanetmath abelian extension of exponent n. If L is the maximal such extensionPlanetmathPlanetmathPlanetmathPlanetmath, choose bL. Then K(b) is a finite extension of exponent dividing n and thus K(b) is of the required form. Thus L=bLK(b) is also of the required form; for example, if SK* is a set of coset representatives for K*/(K*)n, then L=K(S). ∎


Title characterization of abelian extensions of exponent n
Canonical name CharacterizationOfAbelianExtensionsOfExponentN
Date of creation 2013-03-22 18:42:13
Last modified on 2013-03-22 18:42:13
Owner rm50 (10146)
Last modified by rm50 (10146)
Numerical id 4
Author rm50 (10146)
Entry type Theorem
Classification msc 12F10