# OhioLINKElectronic Theses & Dissertations Center

Search ETDs:
The Hasse-Minkowski Theorem in Two and Three Variables
Hoehner, Steven D.

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.

# APA Citation

Hoehner, S. (2012). The Hasse-Minkowski Theorem in Two and Three Variables. (Electronic Thesis or Dissertation). Retrieved from https://etd.ohiolink.edu/

# MLA Citation

Hoehner, Steven. "The Hasse-Minkowski Theorem in Two and Three Variables." Electronic Thesis or Dissertation. Ohio State University, 2012. OhioLINK Electronic Theses and Dissertations Center. 20 Jan 2018.

# Chicago Citation

Hoehner, Steven "The Hasse-Minkowski Theorem in Two and Three Variables." Electronic Thesis or Dissertation. Ohio State University, 2012. https://etd.ohiolink.edu/