{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/86782"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/86782","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Ideal Membership in Polynomial Rings Over the Integers","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.","abstract_html":"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.","abstract_has_math":false,"creators":["Aschenbrenner, Matthias"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["van den Dries, Lou"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2015,"date_issued":"2015-09-28T15:19:30Z","date_published":"2015-09-28T15:19:30Z","updated_at":"2026-07-22T22:26:27Z","subjects":["Mathematics"],"languages":["eng"],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["(MiAaPQ)AAI3023011"],"render_values":[{"text":"(MiAaPQ)AAI3023011","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/86782","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["van den Dries, Lou"]},{"key":"dc:creator","label":"Author","values":["Aschenbrenner, Matthias"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2015-09-28T15:19:30Z","10000-01-01","2001"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["eng"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/86782","(MiAaPQ)AAI3023011"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["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). 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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). 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