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Dissertations / Theses on the topic 'Quaternions algebra'

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Published: 7 June 2025

Last updated: 16 July 2025

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1

GÜNAŞTI, Gökmen. "Quaternions Algebra, Their applications in Rotations and Beyond Quaternions." Thesis, Linnéuniversitetet, Institutionen för datavetenskap, fysik och matematik, DFM, 2012. http://urn.kb.se/resolve?urn=urn:nbn:se:lnu:diva-20267.

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The theory of quaternions was discovered in the middle of nineteenth century and they were commonly used to represent rotations.This thesis is written to review the basic properties of quaternions algebra and their applications in representing rotation of a body in 3-dimensional Euclidean space. Also, last sections in this thesis explore why the use of quaternions are more advantages than Euler angle sequences and can quaternions themselves be further generalized to another number systems?

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2

Lopes, Wilder Bezerra. "Geometric-algebra adaptive filters." Universidade de São Paulo, 2016. http://www.teses.usp.br/teses/disponiveis/3/3142/tde-22092016-143525/.

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This document introduces a new class of adaptive filters, namely Geometric- Algebra Adaptive Filters (GAAFs). Those are generated by formulating the underlying minimization problem (a least-squares cost function) from the perspective of Geometric Algebra (GA), a comprehensive mathematical language well-suited for the description of geometric transformations. Also, differently from the usual linear algebra approach, Geometric Calculus (the extension of Geometric Algebra to differential calculus) allows to apply the same derivation techniques regardless of the type (subalgebra) of the data, i.e.

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Rodríguez, Ordóñez Hugo. "Topological study of nonsingular bilinear maps /." view abstract or download file of text, 2006. http://proquest.umi.com/pqdweb?did=1251841791&sid=5&Fmt=2&clientId=11238&RQT=309&VName=PQD.

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Thesis (Ph. D.)--University of Oregon, 2006.<br>Typescript. Includes vita and abstract. Includes bibliographical references (leaves - ). Also available for download via the World Wide Web; free to University of Oregon users.

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4

Silva, Julio César Conegundes da 1986. "G2 e as álgebras normadas." [s.n.], 2012. http://repositorio.unicamp.br/jspui/handle/REPOSIP/305803.

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Orientador: Luiz Antonio Barrera San Martin<br>Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica<br>Made available in DSpace on 2018-08-21T14:02:28Z (GMT). No. of bitstreams: 1 Silva_JulioCesarConegundesda_M.pdf: 1171316 bytes, checksum: 4e6e6eb2a3f1c066ac73e86495c06428 (MD5) Previous issue date: 2012<br>Resumo: ...Observações: Por apresentar basicamente fórmulas, o resumo, na íntegra, poderá ser visualizado no texto completo da tese digital<br>Abstract: ...Note: The complete abstract is available with the full electronic

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5

Floderová, Hana. "Geometrické struktury založené na kvaternionech." Master's thesis, Vysoké učení technické v Brně. Fakulta strojního inženýrství, 2010. http://www.nusl.cz/ntk/nusl-229021.

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A pair (V, G) is called geometric structure, where V is a vector space and G is a subgroup GL(V), which is a set of transmission matrices. In this thesis we classify structures, which are based on properties of quaternions. Geometric structures based on quaternions are called triple structures. Triple structures are four structures with similar properties as quaternions. Quaternions are generated from real numbers and three complex units. We write quaternions in this shape a+bi+cj+dk.

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6

Freitas, José Roberto. "Equações algébricas nos quatérnios de Hamilton." Universidade Tecnológica Federal do Paraná, 2013. http://repositorio.utfpr.edu.br/jspui/handle/1/594.

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Capes<br>A descoberta dos quatérnios pelo matemático britânico William Rowan Hamilton (1805-1865) permitiu uma nova abordagem na resolução de equações algébricas, fornecendo uma estrutura algébrica mais geral onde buscar soluções. Generalizando o caso clássico (sobre os complexos) apresentamos neste trabalho um tratamento da equação algébrica geral com coeficientes quatérnios. Verificamos que o número de raízes pode ser maior que o grau, e muitas vezes, pode mesmo ser infinito. Damos ênfase ao caso da equação quadrática, obtendo fórmulas para as raízes. Também nos detemos na obtenção de uma ra

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7

Resende, Adriana Souza. "Introdução elementar às álgebras Clifford 'CL IND.2' 'CL IND. 3'." [s.n.], 2010. http://repositorio.unicamp.br/jspui/handle/REPOSIP/306698.

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Orientador: Waldyr Alves Rodrigues Junior<br>Dissertação (mestrado profissional) - Universidade Estadual de Campinas, Instituto de Matemática, Estatistica e Computação Cientifica<br>Made available in DSpace on 2018-08-15T23:09:32Z (GMT). No. of bitstreams: 1 Resende_AdrianaSouza_M.pdf: 17553204 bytes, checksum: a66cefe30e9957cc4351e03d3aec35b2 (MD5) Previous issue date: 2010<br>Resumo: O presente trabalho tem a intenção de apresentar por intermédio de uma linguagem unificada alguns conceitos de cálculo vetorial, álgebra linear (matrizes e transformações lineares) e também algumas idéias elem

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8

Vieira, Vandenberg Lopes. "Grupos fuchsianos aritmeticos identificados em ordens dos quaternios para construção de constelações de sinais." [s.n.], 2007. http://repositorio.unicamp.br/jspui/handle/REPOSIP/261079.

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Orientadores: Reginaldo Palazzo Jr., Mercio Botelho Faria<br>Tese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Eletrica e de Computação<br>Made available in DSpace on 2018-08-08T06:25:10Z (GMT). No. of bitstreams: 1 Vieira_VandenbergLopes_D.pdf: 990187 bytes, checksum: 2212b8074f5503f78aa813ce4422cc4b (MD5) Previous issue date: 2007<br>Resumo: Dentro do contexto de projetar sistema de comunicação digital em espaços homogêneos, em particular, em espaços hiperbólicos, é necessário estabelecer um procedimento sistemático para construção de reticulados O, como element

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9

Parcollet, Titouan. "Quaternion neural networks A survey of quaternion neural networks - Chapter 2 Real to H-space Autoencoders for Theme Identification in Telephone Conversations - Chapter 7." Thesis, Avignon, 2019. http://www.theses.fr/2019AVIG0233.

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Au cours des dernières années, l’apprentissage profond est devenu l’approche privilégiée pour le développement d’une intelligence artificielle moderne (IA). L’augmentation importante de la puissance de calcul, ainsi que la quantité sans cesse croissante de données disponibles ont fait des réseaux de neurones profonds la solution la plus performante pour la resolution de problèmes complexes. Cependant, la capacité à parfaitement représenter la multidimensionalité des données réelles reste un défi majeur pour les architectures neuronales artificielles.Pour résoudre ce problème, les réseaux de ne

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10

Lesesvre, Didier. "Arithmetic Statistics for Quaternion Algebras." Thesis, Sorbonne Paris Cité, 2018. http://www.theses.fr/2018USPCD040.

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Les formes automorphes sont des objets centraux en théorie des nombres. En dépit de leur omniprésence, elles demeurent mystérieuses et leur comportement est loin d'être entièrement compris. Considérer ces formes automorphes au sein de familles a un effet régularisant, et ouvre la voie aux résultats en moyenne : voilà l'esprit des statistiques arithmétiques. La famille de toutes les représentations automorphes d'un groupe réductif donné, appelée famille universelle du groupe, est particulièrement importante. Dans le cas des formes intérieures de GL(2), autrement dit les groupes d'unités d'algèb

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11

Almutairi, Najat Bandar. "ON MULTILINEAR POLYNOMIALS EVALUATED ON QUATERNION ALGEBRA." Kent State University / OhioLINK, 2016. http://rave.ohiolink.edu/etdc/view?acc_num=kent1454279474.

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12

Masson, Simon. "Algorithmique des courbes destinées au contexte de la cryptographie bilinéaire et post-quantique." Electronic Thesis or Diss., Université de Lorraine, 2020. http://www.theses.fr/2020LORR0151.

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Cette thèse étudie l'algorithmie de plusieurs applications cryptographiques liées aux courbes elliptiques et aux isogénies de courbes elliptiques. D'une part, nous étudions le compromis entre efficacité et sécurité concernant les courbes à couplages pour un niveau de sécurité de "128" bits. La menace des récentes avancées sur le calcul de logarithme discret dans certains corps finis nous oriente vers l'étude de nouvelles courbes à couplage. Nous effectuons une comparaison de l'efficacité de ces nouvelles courbes avec celles utilisées actuellement en estimant le temps de calcul pratique. D'autr

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13

Oliveira, Sóstenes Souza de. "R-álgebras de dimensão finita." Universidade Federal de Sergipe, 2017. https://ri.ufs.br/handle/riufs/6521.

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Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES<br>In this work we study the notion of R-algebra. Roughly, they are structures that generalize some arithmetic properties of the body of complex numbers. The ?exibi- lity in this generalization is the non-requirement of properties such as commutativity, associativity and identity element existence. We focus primarily on the ?nite dimen- sional division R-algebras. As is well known, modulo isomorphisms exist exactly four of those R-algebras. In the development of the dissertation we will discuss in detail its main algebra

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14

Whelchel, Pamela Jean. "Analysis on a hyperplane of the quaternions." CSUSB ScholarWorks, 1995. https://scholarworks.lib.csusb.edu/etd-project/1221.

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15

Barton, Christine H. "Magic squares of Lie algebras." Thesis, University of York, 2000. http://etheses.whiterose.ac.uk/10884/.

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16

Déchène, Isabelle. "Quaternion algebras and the graph method for elliptic curves." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1999. http://www.collectionscanada.ca/obj/s4/f2/dsk1/tape9/PQDD_0024/MQ50750.pdf.

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Déchène, Isabelle. "Quaternion algebras and the graph method for elliptic curves." Thesis, McGill University, 1998. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=21537.

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The graph method simultaneously uses the theory of quaternion algebras, elliptic curves and modular forms in order to determine all supersingular points in a given characteristic and hence to obtain a basis of S2(N). The goal of this thesis is to expose the principles of the graph method: it is therefore divided into two main parts: First, we introduce the essentials of the arithmetic of quaternions. This part is made to fit two needs: on one hand, a good introduction or novices; on the other hand, a fast and quick reference for those who are already familiar with the subject. The second part

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Daghigh, Hassan. "Modular forms, quaternion algebras, and special values of L-functions." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1998. http://www.collectionscanada.ca/obj/s4/f2/dsk1/tape11/PQDD_0006/NQ44398.pdf.

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Daghigh, Hassan. "Modular forms, quaternion algebras, and special values of L-functions." Thesis, McGill University, 1997. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=34938.

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Let f be a cusp form of weight 2 and level N. Let K be an imaginary quadratic field of discriminant, --D, and A an ideal class of K. We obtain precise formulas for the special values of the L-functions associated to the Rankin convolution of f and a theta series associated to the ideal class A , in terms of the Petersson scalar product of f with the theta series associated to an Eichler order in a positive definite quaternion algebra. Our work is an extension of the work done by Gross [7]. The central tools used in this thesis are Rankin's method and a reformulation of Gross of work of Waldspu

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Pukhtaievych, Roman. "Periodic and hypercomplex potentials. Properties and applications." Doctoral thesis, Università degli studi di Padova, 2018. http://hdl.handle.net/11577/3425759.

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This Dissertation is devoted to the study of boundary value problems and concerns two research areas. The first one is related to the perturbation analysis of boundary value problems in perforated domains and its application to the investigation of effective properties of composite materials. We investigate the dependence of the solutions of transmission boundary value problems upon some parameters and their behavior when the parameter corresponding to the size of the inclusions tends to zero, and the other parameters tend to some fixed values. Then we apply our results to study the effecti

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Bushman, Nathan. "Hypercomplex Numbers and Early Vector Systems: A History." The Ohio State University, 2020. http://rave.ohiolink.edu/etdc/view?acc_num=osu1585666516546138.

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22

Figueredo, Luis Felipe da Cruz. "Kinematic control based on dual quaternion algebra and its application to robot manipulators." reponame:Repositório Institucional da UnB, 2016. http://repositorio.unb.br/handle/10482/21834.

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Tese (doutorado)—Universidade de Brasília, Faculdade de Tecnologia, Departamento de Engenharia Elétrica, 2016.<br>Submitted by Fernanda Percia França (fernandafranca@bce.unb.br) on 2016-09-23T17:09:08Z No. of bitstreams: 1 2016_LuisFelipedaCruzFigueredo.pdf: 42251540 bytes, checksum: bf57843e92b37b40c0684528bcf64a23 (MD5)<br>Approved for entry into archive by Raquel Viana(raquelviana@bce.unb.br) on 2016-11-24T13:58:06Z (GMT) No. of bitstreams: 1 2016_LuisFelipedaCruzFigueredo.pdf: 42251540 bytes, checksum: bf57843e92b37b40c0684528bcf64a23 (MD5)<br>Made available in DSpace on 2016-11-24T13:58:

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Runge, Piotr. "A Comparison Theorem for the Topological and Algebraic Classification of Quaternionic Toric 8-Manifolds." DigitalCommons@USU, 2009. https://digitalcommons.usu.edu/etd/501.

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In order to discuss topological properties of quaternionic toric 8-manifolds, we introduce the notion of an algebraic morphism in the category of toric spaces. We show that the classification of quaternionic toric 8-manifolds with respect to an algebraic isomorphism is finer than the oriented topological classification. We construct infinite families of quaternionic toric 8-manifolds in the same oriented homeomorphism type but algebraically distinct. To prove that the elements within each family are of the same oriented homeomorphism type, and that we have representatives of all such types of

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Santos, Clotilzio Moreira dos. "Sobre corpos n-hilbertianos e aneis de Witt de algebras de quaternios com divisão." [s.n.], 1994. http://repositorio.unicamp.br/jspui/handle/REPOSIP/306630.

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Orientador: Antonio Paques<br>Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Científica<br>Made available in DSpace on 2018-07-19T17:39:17Z (GMT). No. of bitstreams: 1 Santos_ClotilzioMoreirados_D.pdf: 2784105 bytes, checksum: 8e7a43bb148844822472d9fdcef119b8 (MD5) Previous issue date: 1994<br>Resumo: Não informado.<br>Abstract: Not informed.<br>Doutorado<br>Doutor em Ciências

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Horníček, Jan. "Diskrétně normované řády kvaternionových algeber." Master's thesis, Vysoké učení technické v Brně. Fakulta strojního inženýrství, 2014. http://www.nusl.cz/ntk/nusl-231141.

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Tato práce shrnuje autorův výzkum v oblasti teorie kvaternionových algeber, jejich izomorfismů a maximálních řádů. Nový úhel pohledu na tuto problematiku je umožněn využitím pojmu diskrétní normy. Za hlavní výsledky práce je možná považovat důkaz jednoznačnosti diskrétní normy pro celá čísla, kvadratická rozšíření těles a řády kvaternionových algeber. Dále větu, která umožňuje mezi dvěma kvaternionovými algebrami konstruovat izomorfismy explicitně vyjádřené v maticovém tvaru. A v neposlední řadě důkaz existence nekonečně mnoha různých maximálních řádů kvaternionové algebry. Výsledky uvedené v

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Reynolds, Paul. "On conformal submersions and manifolds with exceptional structure groups." Thesis, University of Edinburgh, 2012. http://hdl.handle.net/1842/6218.

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This thesis comes in three main parts. In the first of these (comprising chapters 2 - 6), the basic theory of Riemannian and conformal submersions is described and the relevant geometric machinery explained. The necessary Clifford algebra is established and applied to understand the relationship between the spinor bundles of the base, the fibres and the total space of a submersion. O'Neill-type formulae relating the covariant derivatives of spinor fields on the base and fibres to the corresponding spinor field on the total space are derived. From these, formulae for the Dirac operators are obt

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Filho, Antonio Calixto de Souza. "Sobre uma classificação dos anéis de inteiros, dos semigrupos finitos e dos RA-loops com a propriedade hiperbólica." Universidade de São Paulo, 2006. http://www.teses.usp.br/teses/disponiveis/45/45131/tde-30012009-163028/.

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Apresentamos duas construções para unidades de uma ordem em uma classe de álgebras de quatérnios que é anel de divisão: as unidades de Pell e as unidades de Gauss. Classificamos os anéis de inteiros de extensões quadráticas racionais, $R$, cujo grupo de unidades $\\U (R G)$ é hiperbólico para um certo grupo $G$ fixado. Também classificamos os semigrupos finitos $S$, tal que, para a álgebra unitária $\\Q S$ e para toda $\\Z$-ordem $\\Gamma$ de $\\Q S$, o grupo de unidades $\\U (\\Gamma)$ é hiperbólico. Nesse mesmo contexto, classificamos os {\\it RA}-loops $L$ cujo loop de unidades $\\U (\\Z L)

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Munhoz, Marcos. "Tópicos de álgebras alternativas." Universidade de São Paulo, 2007. http://www.teses.usp.br/teses/disponiveis/45/45131/tde-16052007-021736/.

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São estudados alguns aspectos das álgebras alternativas, como o bar-radical de uma álgebra bárica alternativa e as identidades de grau 4 e 5 nas álgebras de Cayley-Dickson. Neste estudo fazemos uso da decomposição de Peirce e de diversas propriedades importantes das álgebras alternativas. Concluímos mostrando que as únicas identidades de grau 4 são as triviais e as de grau 5 são conseqüência de outras duas identidades conhecidas<br>We studied some aspects of alternative algebras, in special the bar radical of an alternative baric algebra and identities of degree 4 and 5 of Cayley-Dickson algeb

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Pedone, M. (Matteo). "Algebraic methods for constructing blur-invariant operators and their applications." Doctoral thesis, Oulun yliopisto, 2015. http://urn.fi/urn:isbn:9789526208770.

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Abstract Image acquisition devices are always subject to physical limitations that often manifest as distortions in the appearance of the captured image. The most common types of distortions can be divided into two categories: geometric and radiometric distortions. Examples of the latter ones are: changes in brightness, contrast, or illumination, sensor noise and blur. Since image blur can have many different causes, it is usually not convenient and also computationally expensive to develop ad hoc algorithms to correct each specific type of blur. Instead, it is often possible to extract a blur

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Lacoursière, Claude. "Ghosts and machines : regularized variational methods for interactive simulations of multibodies with dry frictional contacts." Doctoral thesis, Umeå University, Computing Science, 2007. http://urn.kb.se/resolve?urn=urn:nbn:se:umu:diva-1143.

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<p>A time-discrete formulation of the variational principle of mechanics is used to provide a consistent theoretical framework for the construction and analysis of low order integration methods. These are applied to mechanical systems subject to mixed constraints and dry frictional contacts and impacts---machines. The framework includes physics motivated constraint regularization and stabilization schemes. This is done by adding potential energy and Rayleigh dissipation terms in the Lagrangian formulation used throughout. These terms explicitly depend on the value of the Lagrange multipliers e

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Covolo, Tiffany. "(Z2)n-Superalgebra and (Z2)n-Supergeometry." Thesis, Lyon 1, 2014. http://www.theses.fr/2014LYO10203.

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La présente thèse porte sur le développement d'une théorie d'algèbre linéaire, de géométrie et d'analyse basée sur les algèbres (Z2)n-commutatives, c'est-à-dire des algèbres (Z2)n-graduées associatives unitaires satisfaisant ab = (-1)ba, pour tout couple d'éléments homogènes a, b de degrés deg(a), deg(b) où <.,.> est le produit scalaire usuel). Cette généralisation de la supergéométrie a de nombreuses applications : en mathématiques (l'algèbre de Deligne des superformes différentielles, l'algèbre des quaternions et les algèbres de Clifford en sont des exemples) et même en physique (parap

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Benedito, Cintya Wink de Oliveira 1985. "Construção de grupos fuchsianos aritméticos provenientes de álgebras dos quatérnios e ordens maximais dos quatérnios associados a reticulados hiperbólicos." [s.n.], 2014. http://repositorio.unicamp.br/jspui/handle/REPOSIP/261092.

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Orientadores: Reginaldo Palazzo Júnior, Cátia Regina de Oliveira Quilles Queiroz<br>Tese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Elétrica e de Computação<br>Made available in DSpace on 2018-08-25T14:53:45Z (GMT). No. of bitstreams: 1 Benedito_CintyaWinkdeOliveira_D.pdf: 1485856 bytes, checksum: 50adbb3cffa1343c4a0cd9b3d7586173 (MD5) Previous issue date: 2014<br>Resumo: Na busca por novos sistemas de comunicações muitos trabalhos têm sido realizados com o objetivo de obter constelações de sinais e códigos geometricamente uniformes no plano hiperbólico. Neste c

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Lewis, Elizabeth Faith. "Peter Guthrie Tait : new insights into aspects of his life and work : and associated topics in the history of mathematics." Thesis, University of St Andrews, 2015. http://hdl.handle.net/10023/6330.

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In this thesis I present new insights into aspects of Peter Guthrie Tait's life and work, derived principally from largely-unexplored primary source material: Tait's scrapbook, the Tait–Maxwell school-book and Tait's pocket notebook. By way of associated historical insights, I also come to discuss the innovative and far-reaching mathematics of the elusive Frenchman, C.-V. Mourey. P. G. Tait (1831–1901) F.R.S.E., Professor of Mathematics at the Queen's College, Belfast (1854–1860) and of Natural Philosophy at the University of Edinburgh (1860–1901), was one of the leading physicists and mathema

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Hsiao, Yu-Zhe, and 蕭毓哲. "Color Image Processing Algorithms by Applying Quaternions and Reduced Biquaternions Algebra." Thesis, 2016. http://ndltd.ncl.edu.tw/handle/85798774561249393845.

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博士<br>國立臺灣大學<br>電信工程學研究所<br>104<br>The objective of this dissertation is to demonstrate how quaternion and reduced biquaternion algebra can be applied to multi-dimensional signal processing, in particular color image processing. Thanks for the development of quaternion algebra, reduced biquaternion (RB) algebra, and modern computer technologies, many color image processing techniques that are considered slow and unrealistic become very popular in recent years. Since a color image has three components (red, green, and blue), we can encode its pixels to quaternions (or RB) and consider the who

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Mennen, Carminda Margaretha. "The algebra and geometry of continued fractions with integer quaternion coefficients." Thesis, 2015. http://hdl.handle.net/10539/17643.

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A dissertation submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of requirements for the degree of Master of Science. 2015.<br>We consider continued fractions with coe cients that are in K, the quaternions. In particular we consider coe cients in the Hurwitz integers H in K. These continued fractions are expressed as compositions of M¨obius maps in M R4 1 that act, by Poincar´e extension, as isometries on H5. This dissertation explores groups of 2 2 matrices over K and two particular determinant type functions acting on these grou

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Kou, Hung-Cheng, and 郭泓呈. "Quaternion Algebras." Thesis, 2014. http://ndltd.ncl.edu.tw/handle/62032330897428312926.

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碩士<br>國立交通大學<br>應用數學系所<br>102<br>The goal of this paper is to classify quaternion algebras over the field of rational numbers explicitly. More precisely, we give an algorithm to compute a pair (a,b) of nonzero integers for each isomorphic class of quaternion algebras.

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Mišlanová, Kristína. "Kvaternionové algebry a jednotky." Master's thesis, 2021. http://www.nusl.cz/ntk/nusl-448277.

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The aim of this work is to study the Hamiltonian quaternions H and quaternion alge- bras. The first two chapters are based on the article Quaternion algebras by K. Conrad and the rest on the book Quaternion algebras by J. Voight. In the beginning, we mainly develop the theory about quaternions, quaternion algebras and study equivalent condi- tions for being a split or non-split quaternion algebra. After that, we also characterize, up to isomorphism, quaternion algebras over several fields such as R, C or Fp. In the third chapter, the thesis deals with orders in quaternion algebras, especially

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Lin, Yun-Xiang, and 林運翔. "Quaternion Algebra over Rational Number Field." Thesis, 2014. http://ndltd.ncl.edu.tw/handle/32656382502042452480.

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碩士<br>國立臺灣大學<br>數學研究所<br>102<br>This thesis not only classify all quaternion algebras over rational number field but also describe the group structure of the Brauer group formed by them. The quaternion algebra over rational number field can be roughly classified into two types: the 2 by 2 matrix algebra and division rings. Since all 2 by 2 matrices are isomorphic, we only need to classify division rings into non-isomorphic classes. We study the group of norms and the local Hilbert symbols and show that there are exactly two isomorphic classes of quaternion algebras over the local field unless

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Zi, Yang Sham. "Quaternion Algebras and Quadratic Forms." Thesis, 2008. http://hdl.handle.net/10012/3656.

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The main goal of this Masters' thesis is to explore isomorphism types of quaternion algebras using the theory of quadratic forms, number theory and algebra. I would also present ways to characterize quaternion algebras, and talk about how quaternion algebras are important in Brauer groups by describing a theorem proved by Merkurjev in 1981.

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Sutton, Taliesin. "Automorphic forms on quaternion algebras and central L-values /." 2006. http://www.library.wisc.edu/databases/connect/dissertations.html.

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Chan, Kai-Cheong. "Digraph Algebras over Discrete Pre-ordered Groups." Thesis, 2013. http://hdl.handle.net/10012/7290.

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This thesis consists of studies in the separate fields of operator algebras and non-associative algebras. Two natural operator algebra structures, A ⊗_max B and A ⊗_min B, exist on the tensor product of two given unital operator algebras A and B. Because of the different properties enjoyed by the two tensor products in connection to dilation theory, it is of interest to know when they coincide (completely isometrically). Motivated by earlier work due to Paulsen and Power, we provide conditions relating an operator algebra B and another family {C_i}_i of operator algebras under which, for any o

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"Quadratic forms, orderings and quaternion algebras over rings with many units." Thesis, 1988. http://hdl.handle.net/10388/etd-05252012-100556.

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The "algebraic" theory of quadratic forms over fields of characteristic â‰ 2 dates back to the 1937 paper of Witt [37]. It was in this paper that the Witt ring of a field was first considered. Thirty years later, Pfister published his work on the structure of the Witt ring [31]-[33]. These papers triggered an enormous growth in the theory of quadratic forms. Both [15] and [35] are excellent references for the field case. By the early 1970's, much of the algebraic theory of quadratic forms was extended to semilocal rings. (See [3] or [23]. In [3] the characteristic 2 case is also considered.)

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