Submission date: 21 September 2015.

Abstract:

The methods of nonstandard analysis are applied to algebra and number theory. We extend the results of G. L. Cherlin and others on nonstandard Dedekind rings, for example ultraproducts of the ring of integers of number fields. Such rings possess a rich structure and their external ideals are particularly interesting. The prime ideals can be classified using filters on a lattice of finitely generated maximal ideals. We obtain various valuation rings and determine the value groups and the residue class fields. The usual completions of a number field and the rings of adeles can be described as a subquotient of one nonstandard global field. Furthermore, we show that the nonstandard extension of the algebraic closure of a number field can be used to define C_p and a spherical completion.

Mathematics Subject Classification: Primary 11U10, Secondary 13F05

Keywords and phrases:

Full text arXiv 1509.06203: pdf, ps.