**Algebraic number theory** (代数的整数論)

Topics in Mathematical Science IV (数理科学特論 IV), Nagoya University, Fall 2021

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この講義では代数的整数論の入門的内容を取り扱います。この講義は英語で行われます。予備知識としては、代数学の講義で取り扱われる内容を仮定します。受講学生の多くは日本語で行われた代数学の講義を受講していると思いますが、本講義の最初に代数学の知識の復習を行うことで、学生が代数学における英語の語彙を学ぶことができるようにします。また、英語が苦手な学生のために、日本語を少し取り入れた講義ノートを作成していく予定です。

**NUCT course page**if you want to participate in this course <この講義では代数的整数論の入門的内容を取り扱います。この講義は英語で行われます。予備知識としては、代数学の講義で取り扱われる内容を仮定します。受講学生の多くは日本語で行われた代数学の講義を受講していると思いますが、本講義の最初に代数学の知識の復習を行うことで、学生が代数学における英語の語彙を学ぶことができるようにします。また、英語が苦手な学生のために、日本語を少し取り入れた講義ノートを作成していく予定です。

The main objects of study in Algebraic Number Theory are number fields (finite extensions of the field of rational numbers) and their rings of algebraic integers. These concepts generalize the rational numbers and the usual integers. One goal of this lecture is to understand which properties of the integers are still true for these rings of algebraic integers. For example, every positive integer can be uniquely written as a product of primes, i.e., the integers are a unique factorization domain (UFD). We will see that this property is not true for any ring of integers for a number field. For example, in the ring $\mathbb{Z}[\sqrt{-5}]$, which is the ring of integers of the number field $\mathbb{Q}(\sqrt{-5})$, the number $6$ can be written as a product of irreducible elements in two different ways

\[ 6 = 2 \cdot 3 = (1 + \sqrt{-5})(1-\sqrt{-5})\,.\] We will see that this obstruction can be resolved by considering factorizations on the level of ideals instead of numbers. This will lead to the notion of Dedekind domains, in which any nonzero proper ideal factors uniquely into a product of prime ideals.

In the 19th century it became a common technique to gain insight into integer solutions of polynomial equations using rings of algebraic numbers. One of the most famous polynomial equations are the subject of Fermat's Last Theorem, which states that for $n\geq 3$ the equation

\[ x^n + y^n = z^n \] has no integer solution $x,y,z \in \mathbb{Z}$ with $xyz \neq 0$. We will discuss the history of this problem as motivation and try to prove a special case at the end of this lecture. Another example would be the decomposition of primes in the Gaussian integers $\mathbb{Z}[i]$, which detects whether a prime can be written as a sum of two squares (like $5=2^2 + 1^2$) or not (like 7). One of the most fundamental algebraic invariants of a number field is its ideal class group, which measures, among other things, the failure of unique factorization in the ring of integers. One of the fundamental theorems in Algebraic number theory is the finiteness of the ideal class group. We will see that understanding the ideal class group of the number field $\mathbb{Q}(\zeta_p)$, where $\zeta_p=e^{\frac{2\pi i}{p}}$ is a $p$-th root of unity, will lead to a solution of Fermat's Last Theorem for certain primes $p$ (regular primes). The plan of the lecture is to present this proof, which is due to Kummer, and introduce all the necessary ingredients for it. On this journey, we will get insights into number fields and try to understand their structure as much as possible. If time allows, we will also do some explicit calculations using the computer algebra system SageMath.

\[ 6 = 2 \cdot 3 = (1 + \sqrt{-5})(1-\sqrt{-5})\,.\] We will see that this obstruction can be resolved by considering factorizations on the level of ideals instead of numbers. This will lead to the notion of Dedekind domains, in which any nonzero proper ideal factors uniquely into a product of prime ideals.

In the 19th century it became a common technique to gain insight into integer solutions of polynomial equations using rings of algebraic numbers. One of the most famous polynomial equations are the subject of Fermat's Last Theorem, which states that for $n\geq 3$ the equation

\[ x^n + y^n = z^n \] has no integer solution $x,y,z \in \mathbb{Z}$ with $xyz \neq 0$. We will discuss the history of this problem as motivation and try to prove a special case at the end of this lecture. Another example would be the decomposition of primes in the Gaussian integers $\mathbb{Z}[i]$, which detects whether a prime can be written as a sum of two squares (like $5=2^2 + 1^2$) or not (like 7). One of the most fundamental algebraic invariants of a number field is its ideal class group, which measures, among other things, the failure of unique factorization in the ring of integers. One of the fundamental theorems in Algebraic number theory is the finiteness of the ideal class group. We will see that understanding the ideal class group of the number field $\mathbb{Q}(\zeta_p)$, where $\zeta_p=e^{\frac{2\pi i}{p}}$ is a $p$-th root of unity, will lead to a solution of Fermat's Last Theorem for certain primes $p$ (regular primes). The plan of the lecture is to present this proof, which is due to Kummer, and introduce all the necessary ingredients for it. On this journey, we will get insights into number fields and try to understand their structure as much as possible. If time allows, we will also do some explicit calculations using the computer algebra system SageMath.

**Planned content for this course (This is tentative and might change slightly!):**- Motivation: Primes as sum of squares & Gaussian Integers, Fermat's Last Theorem
- Recall basic algebra: Rings, Fields, Field extension, Ideals, ...
- Algebraic numbers, algebraic integers and algebraic number fields
- Integral bases, Dedekind domains
- Lattices & Minkowski Theory
- Ideal class group
- Dirichlet's unit theorem
- Cyclotomic Fields
- Kummers proof of Fermat's Last Theorem for regular primes
- Doing algebraic number theory in SageMath
- Related topics, e.g. finite multiple zeta values

**Material:**

- Course overview
- Homework 1, Homework 2, Homework 3, Homework 4, Homework 5
- Lecture notes (Version 10, 3rd February 2022)
- Handwritten notes: Lecture 1, Lecture 2, Lecture 3, Lecture 4, Lecture 5, Lecture 6, Lecture 7, Lecture 8, Lecture 9, Lecture 10, Lecture 11, Lecture 12, Lecture 13, Lecture 14
- Lecture 15 slides (Review & Sage), Lecture 15 Jupyter Sage notebook

- Additional material:

Please contacte me if you want participate in creating the lecture notes in overleaf.

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**Lecture schedule**

**:**

The lecture will be each

**Friday 10:30 - 12:00.**

**The first lecture will be on 8th October 2021 in the Room 多409 (Math Building).**

It is planned that this lecture will be done face-to-face. If the situation changes we will switch to Zoom lectures.

Week 01 (10/04-10/10): Introduction to Algebraic Number Theory, Gaussian Integers & Ring theory I

Week 02 (10/11-10/17): Gaussian Integers & Ring theory II

Week 03 (10/18-10/24): Ideals

Week 04 (10/25-10/31): Modules & Integrality I

Week 05 (11/01-11/07): Integrality II, Trace & Norm

Week 06 (11/08-11/14): Discriminant

Week 07 (11/15-11/21): Dedekind domains

Week 08 (11/22-11/28): Ideal class group & Lattices

Week 09 (11/29-12/05): Minkowski Theory

Week 10 (12/05-12/12): The Class Number I

Week 11 (12/12-12/19): The Class Number II & Fermat's Last Theorem

Week 12 (12/20-12/26): (There will be no lecture on 24th December 🎅🎄)

**Winter Vacation (12/27-01/07)**☃️

Week 13 (01/10-01/16): Dirichlet's unit theorem

Week 14 (01/17-01/23): Regulator, Extensions of Dedekind domains I

Week 15 (01/24-01/30): Extensions of Dedekind domains II

Week 16 (01/31-02/06): Review & Sage

**References:**

- [C] K. Conrad: "Fermat's last theorem for regular primes", note.
- [IR] K. Ireland and M. Rosen: "A Classical Introduction to Modern Number Theory", 2nd Edition, Springer-Verlag, Berlin.
- [KKS] Kazuya Kato (加藤 和也), Nobushige Kurokawa (黒川 信重 ), Takeshi Saito (斎藤 毅): "数論〈1〉Fermatの夢と類体論 単行本" (Number Theory 1: Fermat's Dream), Tankobon Hardcover 2005/1/7.
- [N] J. Neukirch: "Algebraic Number Theory", Grundlehren der Mathematischen Wissenschaften. 322. Springer-Verlag.

(この本の日本語訳もあります) - [O] T. Ono (小野 孝): "An Introduction to Algebraic Number Theory" translation of the original:「数論序説」.
- [ST] Ian Stewart, David Tall: "Algebraic Number Theory and Fermat's Last Theorem", Chapman and Hall/CRC; 4th edition.

Last update: 4th February 2022