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The Hasse-Minkowski Theorem in Two and Three Variables

Hoehner, Steven D.
http://rave.ohiolink.edu/etdc/view?acc_num=osu1338317481

Abstract Details

2012, Master of Science, Ohio State University, Mathematics.
In this thesis we develop an explanation and proof of the Hasse-Minkowski Theorem for homogeneous quadratic forms in two and three variables using only undergraduate number theory. The goal of this approach is to provide an accessible introduction to this celebrated result of number theory for undergraduates and advanced high school students. Our account of the Hasse-Minkowski Theorem will be expository in nature, providing the reader with the necessary background information to state and prove the theorem. However, some mathematical maturity of the reader is presumed, including familiarity with unique prime factorization, greatest common divisor, least common multiple, the Division Algorithm, the Euclidean Algorithm, the Pigeonhole Principle, and some other basic notions from elementary number theory, set theory, and linear algebra. The remainder of the information necessary for the explanation and proof of the Hasse-Minkowski Theorem is provided in the paper, making it largely self-contained in this respect. Moreover, throughout the paper we provide the reader with several numerical examples of the various concepts and methods introduced. We now provide the reader with a brief overview of the paper. The first chapter consists of a brief historical account of the study of Diophantine equations in number theory, including Hasse and Minkowski’s important contributions to the study of quadratic forms. In Chapter 2 we motivate the discussion of the Hasse-Minkowski Theorem by introducing the type of solutions we seek, and in Chapters 3 and 4 we provide some background information on quadratic forms in two and three variables, respectively. Chapter 5 serves as an introduction to modular arithmetic and some of the results that will be used in the proof of the Hasse-Minkowski Theorem, while in Chapters 6 and 7 we state and prove the Hasse-Minkowski Theorem for forms in two and three variables, respectively. Finally, in Chapter 8 we discuss topics for further study.
C. Herbert Clemens, PhD (Advisor)
James Cogdell, PhD (Committee Member)
46 p.
Mathematics
Hasse; Minkowski; Diophantine; number theory

Recommended Citations

Citations

  • Hoehner, S. D. (2012). The Hasse-Minkowski Theorem in Two and Three Variables [Master's thesis, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu1338317481

    APA Style (7th edition)

  • Hoehner, Steven. The Hasse-Minkowski Theorem in Two and Three Variables. 2012. Ohio State University, Master's thesis. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=osu1338317481.

    MLA Style (8th edition)

  • Hoehner, Steven. "The Hasse-Minkowski Theorem in Two and Three Variables." Master's thesis, Ohio State University, 2012. http://rave.ohiolink.edu/etdc/view?acc_num=osu1338317481

    Chicago Manual of Style (17th edition)

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© 2012, all rights reserved.
This open access ETD is published by The Ohio State University and OhioLINK.