Galois subfields of real radical extensions are at most quadratic
Suppose are fields with and Galois over . Then .
Now, is Galois since is. But is a Kummer extension of , so has cyclic Galois group and thus has cyclic Galois group as well (being a quotient of ). Thus is a Kummer extension of , so that for some . It follows that . But since is Galois over , it follows that (since otherwise in order to be Galois, would have to contain the non-real roots of unity).
|Title||Galois subfields of real radical extensions are at most quadratic|
|Date of creation||2013-03-22 17:43:05|
|Last modified on||2013-03-22 17:43:05|
|Last modified by||rm50 (10146)|