On a theorem of Bauer and some of its applications
Let be a pure number field generated by a complex root of a monic irreducible polynomial , where , , are three positive natural integers. The purpose of this paper is to study the monogenity of . Our results are illustrated by some examples.
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.
Let be a number field generated by a complex root of a monic irreducible polynomial , , is a square free rational integer. We prove that if or and , then the number field is monogenic. If or , then the number field is not monogenic.
Let be an extension of a number field , where satisfies the monic irreducible polynomial of prime degree belonging to ( is the ring of integers of ). The purpose of this paper is to study the monogenity of over by a simple and practical version of Dedekind’s criterion characterizing the existence of power integral bases over an arbitrary Dedekind ring by using the Gauss valuation and the index ideal. As an illustration, we determine an integral basis of a pure nonic field with a...
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...