Ideal Membership in Polynomial Rings Over the Integers

The approach to the ideal membership problem for Z [X] followed here is based on some properties (such as Weierstrass Division) of the ring Z p⟨X⟩ of restricted power series with coefficients in the ring Z p of p-adic integers. We also consider the ideal membership problem for ideals of the ring Z p⟨X⟩ itself, and for ideals of its subring Z p⟨X⟩alg consisting of the restricted p-adic power series which are algebraic over Z [X]. Here, we make extensive use of a height function on the algebraic closure of Q (X) introduced by Kani (1978). Among other things, we obtain an effective version of the Weierstrass Division Theorem for the ring Z p⟨X⟩alg.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2015

Author dc:creator

• Aschenbrenner, Matthias

Contributors dc:contributor

• van den Dries, Lou

• Mathematics

Language dc:language

eng

Identifier

(MiAaPQ)AAI3023011

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/86782