characterization of abelian extensions of exponent n
Theorem 1.
Let K be a field containing the nth roots of unity, with characteristic not dividing n. Let L be a finite extension of K. Then the following are equivalent:
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1.
L/K is Galois with abelian Galois group of exponent dividing n
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2.
L=K(n√a1,…,n√ak) for some ai∈K.
Proof.
Let ζ∈K be a primitive nth root of unity.
(2⇒1): Choose αi∈L such that αni=ai∈K. 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 separable over K and splits in L, so that L is the splitting field of the set of polynomials {xn-ai∣ 1≤i≤k}. Thus L/K is Galois. Given σ∈Gal(L/K), for each i we have σ(αi)=ζjαi for some 1≤j≤n, so that σk(αi)=ζkjαi. It follows that σn is the identity for every σ∈Gal(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 multiplication by some nth root of unity: if σ,τ∈Gal(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.
(1⇒2): Let G=Gal(L/K), and write G=C1×…×Cr where each Ci is cyclic; |Ci|=mi∣n for each i. For each i, define a subgroup Hi≤G by
Hi=C1×…×Ci-1×Ci+1×…×Cr |
Then G/Hi≅Ci. Let Li be the fixed field of Hi. Li is normal over K since Hi is normal in G, and Gal(Li/K)≅G/Hi≅Ci and thus Li/K is cyclic Galois of order mi. K contains the primitive mthi root of unity ζn/mi and thus Li=K(αi) for some αi∈L with αmii∈K (by Kummer theory). But then also αni∈K. Then
Gal(L:K(α1,…,αr))=H1∩…∩Hr={1} |
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 extension of K of exponent n, where n is prime to the characteristic of K, then L=K({n√a}) for some set of a∈K.
Proof.
Clearly K({n√a∣a∈K*) is an infinite abelian extension of exponent n. If L is the maximal such extension, choose b∈L. Then K(b) is a finite extension of exponent dividing n and thus K(b) is of the required form. Thus L=∪b∈LK(b) is also of the required form; for example, if S⊂K* is a set of coset representatives for K*/(K*)n, then L=K(S). ∎
References
- 1 Morandi, P., Field and Galois Theory, Springer, 1996.
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 |