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van den Dries, Lou Aschenbrenner, Matthias 2015-09-28T15:19:30Z 2015-09-28T15:19:30Z 10000-01-01 2001 2001 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. Made available in DSpace on 2015-09-28T15:19:30Z (GMT). No. of bitstreams: 2 license.txt: 4848 bytes, checksum: 96035ab3f5e1c23cc7138a224ce498bd (MD5) 3023011.pdf: 8056315 bytes, checksum: 79f70e638eb5bc50fd1b5fb249ed1e7b (MD5) Previous issue date: 2001 Embargo set by: Seth Robbins for item 88063 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs U of I Only 181 p. Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 2001. http://hdl.handle.net/2142/86782 (MiAaPQ)AAI3023011 eng Mathematics Ideal Membership in Polynomial Rings Over the Integers text Mathematics Mathematics University of Illinois at Urbana-Champaign Dissertation Ph.D.