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 Title: Ideal Membership in Polynomial Rings Over the Integers Author(s): Aschenbrenner, Matthias Doctoral Committee Chair(s): van den Dries, Lou Department / Program: Mathematics Discipline: Mathematics Degree Granting Institution: University of Illinois at Urbana-Champaign Degree: Ph.D. Genre: Dissertation Subject(s): Mathematics 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. Issue Date: 2001 Type: Text Language: English Description: 181 p.Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 2001. URI: http://hdl.handle.net/2142/86782 Other Identifier(s): (MiAaPQ)AAI3023011 Date Available in IDEALS: 2015-09-28 Date Deposited: 2001
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