Quest for Mathematics II-E2

Numbering Code U-LAS10 10024 LE55 Year/Term 2021 ・ First semester
Number of Credits 2 Course Type seminar
Target Year All students Target Student For all majors
Language English Day/Period Thu.4
Instructor name TAN, Fucheng (Research Institute for Mathematical Sciences Senior Lecturer)
Outline and Purpose of the Course You might have heard of the following expression from Gauss (1777-1855): "Mathematics is the queen of sciences and number theory is the queen of mathematics. She often condescends to render service to astronomy and other natural sciences, but in all relations she is entitled to the first rank."

What is number theory? At the most basic level, it is the study of the properties of the integers Z={..., -2, -1, 0, 1, 2, ...}.

In this course, we will study certain topics in elementary number theory, including (but not limited to) divisibility, congruences, quadratic reciprocity, and arithmetic progressions. Some abstract algebra will be introduced in class as a tool of number theory.
Course Goals The class is meant to help students of all disciplines improve their knowledges in number theory. Moreover, students will improve their communication skills in English via oral discussions and presentations.
Schedule and Contents Below is the contents and schedules of the course. Some of these topics may be assigned to the students for their presentations. The lectures and presentations, as well as their orders, may be modified, depending on students' backgrounds and understanding of the course materials. The instructor will provide corrections and comments on students' presentations.

(1) Introduction (Week 1)
-Some basics in set theory and logic, motivating examples and conjectures, remarks on the course materials.

(2) Divisibility (Weeks 2-3)
-The division algorithm, the Euclidean algorithm, prime numbers;
-The fundamental theorem of arithmetic, infinity of the set of prime numbers, the binomial theorem.

(3) Congruences (Weeks 4-6)
-Basic definitions and properties, Fermat's theorem and Euler's generalization, Wilson's theorem;
-The Chinese Remainder theorem, Hensel's lemma;
-Prime (power) moduli, primitive roots.

(4) Quadratic reciprocity (Weeks 7-9)
-Legendre symbols, Lemma of Gauss, the reciprocity law;
-Binary quadratic forms;
-Gaussian integers, two squares theorem.

(5) Arithmetic progression (Week 10-14)
-Characters of finite abelian groups;
-Dirichlet series;
-Riemann Zeta function and L-functions;
-Theorem of arithmetic progressions.

Total:14 classes, 1 Feedback session
Evaluation Methods and Policy The evaluation consists of three weighted parts:

- Discussion performance in class (20%).

- Presentation (60%): Each student reviews a mathematical topic assigned by the instructor.

- Report (20%): An essay on the topic of presentation.
Course Requirements There are no formal prerequisites for the class. Some familiarity with mathematical proofs (e.g. as one sees in Calculus and Linear Algebra) will be helpful, but not required.
Study outside of Class (preparation and review) Along with preparation and review, students are encouraged to form study groups.
Textbooks Textbooks/References Number Theory for Beginners, A. Weil, (Springer), ISBN:9781461299585, E-book available at Kyoto U library
An Introduction to the Theory of Numbers, Ivan Niven, Herbert Zuckerman, and Hugh Montgomery, ( Wiley), ISBN:9780471625469, This book is thorough
A Course in Arithmetic, Jean-Pierre Serre, (Springer), ISBN:9780387900407, E-book available at Kyoto U library
References, etc. A Classical Introduction to Modern Number Theory, Kenneth Ireland and Michael Rosen, (Springer), ISBN:9780387973296, This book may be helpful to the students who have studied modern algebra systematically.
Algebraic Number Theory, J. S. Milne, This lecture note may be helpful to the students who have studied modern algebra systematically.
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