On a theorem of Bauer and some of its applications
We give a necessary condition for a surjective representation Gal to arise from the -torsion of a -curve. We pay a special attention to the case of quadratic -curves.
We study the compositum of all degree extensions of a number field in a fixed algebraic closure. We show contains all subextensions of degree less than if and only if . We prove that for there is no bound on the degree of elements required to generate finite subextensions of . Restricting to Galois subextensions, we prove such a bound does not exist under certain conditions on divisors of , but that one can take when is prime. This question was inspired by work of Bombieri and...