Persistent Link:
http://hdl.handle.net/10150/194074
Title:
On p-adic Continued Fractions and Quadratic Irrationals
Author:
Miller, Justin Thomson
Issue Date:
2007
Publisher:
The University of Arizona.
Rights:
Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.
Abstract:
In this dissertation we investigate prior definitions for p-adic continued fractions and introduce some new definitions. We introduce a continued fraction algorithm for quadratic irrationals, prove periodicity for Q₂ and Q₃, and numerically observe periodicity for Q(p) when p < 37. Various observations and calculations regarding this algorithm are discussed, including a new type of symmetry observed in many of these periods, which is different from the palindromic symmetry observed for real continued fractions and some previously defined p-adic continued fractions. Other results are proved for p-adic continued fractions of various forms. Sufficient criteria are given for a class of p-adic continued fractions of rational numbers to be finite. An algorithm is given which results in a periodic continued fraction of period length one for √D ∈ Zˣ(p), D ∈ Z, D non-square; although, different D require different parameters to be used in the algorithm. And, a connection is made between continued fractions and de Weger’s approximation lattices, so that periodic continued fractions can be generated from a periodic sequence of approximation lattices, for square roots in Zˣ(p). For simple p-adic continued fractions with rational coefficients, we discuss observations and calculations related to Browkin’s continued fraction algorithms. In the last chapter, we apply some of the definitions and techniques developed in the earlier chapters for Q(p) and Z to the t-adic function field case F(q)((t)) and F(q)[t], respectively. We introduce a continued fraction algorithm for quadratic irrationals in F(q)((t)) that always produces periodic continued fractions.
Type:
text; Electronic Dissertation
Keywords:
continued fractions; p-adic fields; local fields; quadratic irrationals
Degree Name:
PhD
Degree Level:
doctoral
Degree Program:
Mathematics; Graduate College
Degree Grantor:
University of Arizona
Committee Chair:
Thakur, Dinesh S.

Full metadata record

DC FieldValue Language
dc.language.isoENen_US
dc.titleOn p-adic Continued Fractions and Quadratic Irrationalsen_US
dc.creatorMiller, Justin Thomsonen_US
dc.contributor.authorMiller, Justin Thomsonen_US
dc.date.issued2007en_US
dc.publisherThe University of Arizona.en_US
dc.rightsCopyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.en_US
dc.description.abstractIn this dissertation we investigate prior definitions for p-adic continued fractions and introduce some new definitions. We introduce a continued fraction algorithm for quadratic irrationals, prove periodicity for Q₂ and Q₃, and numerically observe periodicity for Q(p) when p < 37. Various observations and calculations regarding this algorithm are discussed, including a new type of symmetry observed in many of these periods, which is different from the palindromic symmetry observed for real continued fractions and some previously defined p-adic continued fractions. Other results are proved for p-adic continued fractions of various forms. Sufficient criteria are given for a class of p-adic continued fractions of rational numbers to be finite. An algorithm is given which results in a periodic continued fraction of period length one for √D ∈ Zˣ(p), D ∈ Z, D non-square; although, different D require different parameters to be used in the algorithm. And, a connection is made between continued fractions and de Weger’s approximation lattices, so that periodic continued fractions can be generated from a periodic sequence of approximation lattices, for square roots in Zˣ(p). For simple p-adic continued fractions with rational coefficients, we discuss observations and calculations related to Browkin’s continued fraction algorithms. In the last chapter, we apply some of the definitions and techniques developed in the earlier chapters for Q(p) and Z to the t-adic function field case F(q)((t)) and F(q)[t], respectively. We introduce a continued fraction algorithm for quadratic irrationals in F(q)((t)) that always produces periodic continued fractions.en_US
dc.typetexten_US
dc.typeElectronic Dissertationen_US
dc.subjectcontinued fractionsen_US
dc.subjectp-adic fieldsen_US
dc.subjectlocal fieldsen_US
dc.subjectquadratic irrationalsen_US
thesis.degree.namePhDen_US
thesis.degree.leveldoctoralen_US
thesis.degree.disciplineMathematicsen_US
thesis.degree.disciplineGraduate Collegeen_US
thesis.degree.grantorUniversity of Arizonaen_US
dc.contributor.chairThakur, Dinesh S.en_US
dc.contributor.committeememberMadden, Danielen_US
dc.contributor.committeememberLux, Klausen_US
dc.identifier.proquest2433en_US
dc.identifier.oclc659748350en_US
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