**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

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。

This course will be done in English. The way of teaching (face-to-face or online) is not decided yet. If there are students interested in this course who can not come to the campus (i.e. NUPACE students) I would try to offer parts of this course online. Please contact me if you are interested in this course but you can not come to the campus in October.

This course will be done in English. The way of teaching (face-to-face or online) is not decided yet. If there are students interested in this course who can not come to the campus (i.e. NUPACE students) I would try to offer parts of this course online. Please contact me if you are interested in this course but you can not come to the campus in October.

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
- Dedekind 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:**

- Homework assignments & Lectures notes will appear here -

The current idea is to create summarizing lecture notes together with the participants by using 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.**

The place and format (face-to-face or online) is not decided yet. But I hope that this lecture will be done face-to-face in a classroom. The room will be announced here.

**References:**

**- will be updated in the coming weeks -**

- [IR] K. Ireland and M. Rosen: "A Classical Introduction to Modern Number Theory", 2nd Edition, Springer-Verlag, Berlin.
- [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: 20th July 2021