characterization of abelian extensions of exponent n
Theorem 1.
Let be a field containing the roots of unity, with characteristic
not dividing . Let be a finite extension
of . Then the following are equivalent
:
-
1.
is Galois with abelian
Galois group
of exponent dividing
-
2.
for some .
Proof.
Let be a primitive root of unity.
Choose such that . Then for each , the elements are distinct and are all the roots of in . Thus is separable over and splits in , so that is the splitting field
of the set of polynomials
. Thus is Galois. Given , for each we have for some , so that . It follows that is the identity
for every , so that the exponent of divides . It remains to show that is abelian; this follows trivially from the simple definition of the Galois action as multiplication
by some root of unity: if with , then
Thus on each . But the generate , so on and is abelian.
Let , and write where each is cyclic; for each . For each , define a subgroup by
Then . Let be the fixed field of . is normal over since is normal in , and and thus is cyclic Galois of order . contains the primitive root of unity and thus for some with (by Kummer theory). But then also . Then
since any element of the left-hand group fixes each and thus fixes so is the identity in . Thus . ∎
Corollary 2.
If is the maximal abelian extension of of exponent , where is prime to the characteristic of , then for some set of .
Proof.
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 |