Ideal Membership in Polynomial Rings Over the Integers
Aschenbrenner, Matthias
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https://hdl.handle.net/2142/86782
Description
Title
Ideal Membership in Polynomial Rings Over the Integers
Author(s)
Aschenbrenner, Matthias
Issue Date
2001
Doctoral Committee Chair(s)
van den Dries, Lou
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Mathematics
Language
eng
Abstract
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.
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