characterization of abelian extensions of exponent n
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 . ∎
- 1 Morandi, P., Field and Galois Theory, Springer, 1996.
|Title||characterization of abelian extensions of exponent n|
|Date of creation||2013-03-22 18:42:13|
|Last modified on||2013-03-22 18:42:13|
|Last modified by||rm50 (10146)|