數論專題（一）、2026年秋
Topics in Number Theory (I), Fall 2026
Algebraic Number Theory
Call number: 2406451_01
Time & Place: 二E、四E (13:15--14:30, TuTh), 308A Math. Building
Texts:
[ST16] I. N. Stewart and D. O. Tall, Algebraic Number Theory and Fermat's Last
Theorem, 4th ed., CRC Press, Boca Raton, 2016.
[Ma18] D. A. Marcus, Number Fields, 2nd ed., Springer, Cham, 2018.
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Instructor: Wen-Chin Liaw（廖文欽) Office: 441 Math. Building Phone: 05-2720411 ext. 66126 E-mail: wenchinliaw@gmail.com Office hours: 1--3 Wednesdays (or by appointment) |
Syllabus
We will cover the whole book except for Chapter 1: Algebraic background,
which is assumed as a prerequisite.
Our topics include integral bases, quadratic fields,
cyclotomic fields, Euclidean domains, prime factorizations of ideals, Minkowski's theorem for
lattices, sum of squares, ideal class groups, class numbers,
Fermat's Last Theorem (for regular primes), and Dirichlet's Unit Theorem.
If time permits, we can also discuss fundamental units of real quadratic fileds,
Dirichlet L-functions, and quadratic reciprocity.
Grading Policy and Rules
| 7 | Homework | (due on Thursdays, biweekly) | 70% |
| 1 | Final | (due by Thursday, 2026/12/31) | 30% |
- Assigned homework is due every other Thursday, unless otherwise stated. They are collected at the beginning of the lecture.
- No late homework will be accepted, except in truly extraordinary circumstances.
- There will be no midterm exam.
- A take-home final will be handed out in mid December and should be handed in no later than the last day of class, Thursday, December 31, 2026.
[CF67] J. W. S. Cassels and A. Frohlich, eds., Algebraic Number Theory, Thompson, Washington, D.C., 1967.
[FT93] A. Frohlich and M. J. Taylor, Algebraic Number Theory, Cambridge Univ. Press, Cambridge, UK, 1993.
[La94] S. Lang, Algebraic Number Theory, 2nd ed., Springer-Verlag, New York, 1994.
[Ma18] D. A. Marcus, Number Fields, 2nd ed., Springer, Cham, 2018.
Other related references
[IR90] K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory,
2nd ed., Springer-Verlag, New York, 1990.
[Ja96] G. J. Janusz, Algebraic Number Fields, 2nd ed., Amer. Math. Soc., 1996.
[SO85] W. Scharlau and H. Opolka, From Fermat to Minkowski, Springer-Verlag,
New York, 1985.