Math 530, Algebraic Number Theory
Spring 2026
Time and Place: MWF 2:00-2:50pm in 336 Davenport Hall.
Instructor: William Chen
Email: wyc@illinois.edu
Office: 51 CAB (Computing Applications Building)
Office Hours: Tuesdays 3-4pm, Thursdays 1-2pm
Textbook: James Milne, Algebraic Number Theory(v3.08)
Supplemental text: Marcus, Number Fields
Course Description
This is a graduate course in algebraic number theory. We will develop the theory of rings of integers of number fields. Some topics we will cover include: class groups and finiteness of the class number, Dirichlet’s unit theorem, cyclotomic extensions and applications to Fermat’s Last Theorem. Other potential topics include local fields, global fields, Dedekind zeta functions and the class number formula.
Course Policies
Grading: Your course grade will be based on homework (40%), one midterm exam (30%), and a final exam (30%)
Weekly homework: These will typically be due on Friday before class, to be submitted via gradescope. Late homework will not be accepted, but your lowest two homework scores will be dropped. Collaboration on homework is encouraged. However, you must write up your solutions individually and understand them completely.
In-class midterms: One 50-minute exam to be held in our usual classroom.
Final exam: TBA
Missed exams: There will be no makeup exams. If your circumstances are truly extraordinary, I may excuse you from an exam, in which case your average will be determined by your other exams and homeworks.
Cheating: No.
Disabilities: Students with disabilities who require reasonable accommodations should see me as soon as possible. In particular, any accommodation on exams must be requested at least a week in advance and will require a letter from DRES.
Schedule
Jan 21 - Introduction. The Gaussian integers, primes as sums of 2 squares. [Notes]
Jan 23 - Finite and integral ring maps, integral closure, rings of integers. [Notes]
Jan 26 - More on integral closure. Norm and trace. [Notes]
Jan 28 - More on Norm and trace. Bilinear forms, discriminants. [Notes]
Jan 30 - Nondegeneracy of trace. Finite generation of the ring of integers. [Notes][Problem Set 1] due. [pset 1 solutions]
Feb 2 - Dual bases, computing discriminants. [Notes]
Feb 4 - Computing rings of integers, Stickelberger’s theorem. [Notes]
Feb 6 - Discrete valuation rings [Notes]
Feb 9 - Dedekind domainds [Notes][Problem Set 2] due. [pset2 solutions]
Feb 11 - Ring-space duality. Unique factorization into prime ideals. [Notes]
Feb 13 - Applications of unique factorization. [Problem Set 3] due. [Notes][pset 3 solutions]
Feb 16 - The ideal class group. Rings of integers are Dedekind. [Notes]
Feb 18 - Factorization of primes in extensions. [Notes]
Feb 20 - Ramification. [Problem Set 4] due. [Notes][pset4solutions]
Feb 23 - Computing factorizations in simple extensions. [Notes]
Feb 25 - Discrete valuations. [Notes]
Feb 27 - Eisenstein extensions, Newton polygon. [Problem Set 5] due. [Notes][pset5solutions]
Mar 2 - Ideal norm. [Notes]
Mar 4 - Numerical norm, Minkowski’s bound and the finiteness of the class number. [Notes]
Mar 6 - Lattices. [Problem Set 6] due. [Notes][pset6solutions]
Mar 9 - Minkow'ski’s bound for convex symmetric domains. Four squares theorem. [Notes]
Mar 11 - Some calculus, finiteness of the class number? [Notes]
Mar 13 - Midterm Exam [Problem Set 7] due. [pset7solutions][Exam1solutions]
<Spring break>
Mar 23 - Remarks on Minkowski’s bound. The unit theorem. Statement and preliminaries. [Notes]
Mar 25 - Finite generation of the unit group. [Notes]
Mar 27 - Proof of the unit theorem. [Problem Set 8] due. [pset8solutions][Notes]
Mar 30 - S-integers and the unit theorem for S-units. Applications to cubic fields. [Notes]
Apr 1 - Cyclotomic fields [Notes]
Apr 3 - Fermat’s last theorem for regular primes. [Notes]
Apr 6 - Absolute values [Notes][Problem Set 9] due. [pset9solutions]
Apr 8 - Ostrowski’s theorem. [Notes]
Apr 10 - The weak approximation theorem, completions. [Notes]
April 13 - More on completions, p-adic numbers (Yuan Liu guest lecture).
April 15 - Topology of the p-adic numbers. Hensel’s lemma I. [Notes]
April 17 - Hensels lemma II, extending absolute values. [Notes]
April 20 - Local fields, unramified extensions. [Notes]
April 22 - Ramification.
April 24 - [Problem Set 10] due.
…
May 11 - Final Exam: 7-10pm, 336 Davenport Hall.