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The Geometry of Imaginary Quadratic Fields.

Record Type:	Language materials, manuscript : Monograph/item
Title/Author:	The Geometry of Imaginary Quadratic Fields./
Author:	Martin, Daniel E.
Description:	1 online resource (86 pages)
Notes:	Source: Dissertations Abstracts International, Volume: 81-11, Section: B.
Contained By:	Dissertations Abstracts International81-11B.
Subject:	Mathematics. -
Online resource:	click for full text (PQDT)
ISBN:	9798645450434

The Geometry of Imaginary Quadratic Fields.
Martin, Daniel E.

The Geometry of Imaginary Quadratic Fields. - 1 online resource (86 pages)

Source: Dissertations Abstracts International, Volume: 81-11, Section: B.

Thesis (Ph.D.)--University of Colorado at Boulder, 2020.

Includes bibliographical references

Aspects of an imaginary quadratic field, K, including the class group, associated matrix groups, and Diophantine approximation properties, are studied through the lens of a geometric object called the Schmidt arrangement. It is defined as the orbit of the real line under Mobius transformations associated to SL2(O), where O is the ring of integers. The special linear group is replaced here with more general matrix sets in Chapter 4. By determining which sets correspond to connected arrangements, we are led to new pseudo-Euclidean and continued fraction algorithms.In Chapter 3 we give a fourth proof of Cohn's theorem---that SL2(O) and GL2(O) are generated by their elementary matrices if and only if O is Euclidean. Following Stange, this is reinterpreted as a lack of connectivity in the non-Euclidean Schmidt arrangements, which is addressed by the search for more general arrangements that are connected in Chapter 4.A new pseudo-Euclidean algorithm is introduced in Chapter 5, which manifests as a walk across a connected arrangement. Among its consequences are natural analogs of the statement, "Euclidean implies principal ideal domain," as well as a bound on integers that multiplicatively generate a set, S, that makes the ring of S-integers Euclidean.Chapter 6 rephrases the pseudo-Euclidean algorithm as a generalization of the Hurwitz continued fraction algorithm. We prove that the resulting K-rational approximations to a complex number satisfy many of the properties possessed by the classical continued fractions produced over Q. These properties include exponential convergence, being best approximations of the second kind up to constants, bad approximability equating bounded coefficients, and quadratic irrationality equating periodic coefficients.

Electronic reproduction.
Ann Arbor, Mich. :
ProQuest,
2024

Mode of access: World Wide Web

ISBN: 9798645450434Subjects--Topical Terms:

527692
Mathematics.
Subjects--Index Terms:

Class groupIndex Terms--Genre/Form:

554714
Electronic books.

The Geometry of Imaginary Quadratic Fields.
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500 $a Source: Dissertations Abstracts International, Volume: 81-11, Section: B.
500 $a Advisor: Stange, Katherine.
502 $a Thesis (Ph.D.)--University of Colorado at Boulder, 2020.
504 $a Includes bibliographical references
520 $a Aspects of an imaginary quadratic field, K, including the class group, associated matrix groups, and Diophantine approximation properties, are studied through the lens of a geometric object called the Schmidt arrangement. It is defined as the orbit of the real line under Mobius transformations associated to SL2(O), where O is the ring of integers. The special linear group is replaced here with more general matrix sets in Chapter 4. By determining which sets correspond to connected arrangements, we are led to new pseudo-Euclidean and continued fraction algorithms.In Chapter 3 we give a fourth proof of Cohn's theorem---that SL2(O) and GL2(O) are generated by their elementary matrices if and only if O is Euclidean. Following Stange, this is reinterpreted as a lack of connectivity in the non-Euclidean Schmidt arrangements, which is addressed by the search for more general arrangements that are connected in Chapter 4.A new pseudo-Euclidean algorithm is introduced in Chapter 5, which manifests as a walk across a connected arrangement. Among its consequences are natural analogs of the statement, "Euclidean implies principal ideal domain," as well as a bound on integers that multiplicatively generate a set, S, that makes the ring of S-integers Euclidean.Chapter 6 rephrases the pseudo-Euclidean algorithm as a generalization of the Hurwitz continued fraction algorithm. We prove that the resulting K-rational approximations to a complex number satisfy many of the properties possessed by the classical continued fractions produced over Q. These properties include exponential convergence, being best approximations of the second kind up to constants, bad approximability equating bounded coefficients, and quadratic irrationality equating periodic coefficients.
533 $a Electronic reproduction. $b Ann Arbor, Mich. : $c ProQuest, $d 2024
538 $a Mode of access: World Wide Web
650 4 $a Mathematics. $3 527692
653 $a Class group
653 $a Continued fractions
653 $a Diophantine approximation
653 $a Euclidean algorithm
653 $a Imaginary quadratic field
653 $a Mobius transformations
655 7 $a Electronic books. $2 local $3 554714
690 $a 0405
710 2 $a ProQuest Information and Learning Co. $3 1178819
710 2 $a University of Colorado at Boulder. $b Mathematics. $3 1477632
773 0 $t Dissertations Abstracts International $g 81-11B.
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