授業の目的 【日本語】
Goals of the Course(JPN) | | | 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。 |
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授業の目的 【英語】
Goals of the Course | | All information on this course are regularly updated on the course homepage:
https://www.henrikbachmann.com/algnt_2021.html .
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. 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. We will discuss the history of this problem as motivation and try to prove a special case at the end of this lecture. 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. 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. |
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到達目標 【日本語】
Objectives of the Course(JPN)) | | |
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到達目標 【英語】
Objectives of the Course | | | The goal of this course is to learn the basic concepts of algebraic number theory (See Course purpose for more details). I also hope to encourage Japanese students to use this course to improve their english capabilities to be competitive on an international level which is necessary if one is interested in becoming a researcher in mathematics. |
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授業の内容や構成
Course Content / Plan | | We will start with a motivation on primes as sum of squares & Gaussian Integers and Fermat's Last Theorem. Then we will recall basic algebra such as Rings, Fields, Field extension, Ideals, etc.
Then we will start talking about the main object, namely algebraic number fields for which we will cover all the topics mentioned at "Keywords". |
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履修条件
Course Prerequisites | | | A standard course in Algebra would be recommended. The basics will be recalled at the beginning of the course. |
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関連する科目
Related Courses | | |
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成績評価の方法と基準
Course Evaluation Method and Criteria | | | Grading is based on homework assignment. |
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教科書・テキスト
Textbook | | We will give a list of references at the beginning of the course. The main reference will be:
J. Neukirch: "Algebraic Number Theory", Grundlehren der Mathematischen Wissenschaften. 322. Springer-Verlag. The plan is to also create (together with the participants) summarizing lecture notes in English and Japanese. |
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参考書
Reference Book | | J. Neukirch: "Algebraic Number Theory", Grundlehren der Mathematischen Wissenschaften. 322. Springer-Verlag,
K. Ireland and M. Rosen: "A Classical Introduction to Modern Number Theory", 2nd Edition, Springer-Verlag, Berlin,
Ian Stewart, David Tall: "Algebraic Number Theory and Fermat's Last Theorem", Chapman and Hall/CRC; 4th edition. |
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課外学習等（授業時間外学習の指示）
Study Load(Self-directed Learning Outside Course Hours) | | |
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注意事項
Notice for Students | | | 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。 |
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他学科聴講の可否
Propriety of Other department student's attendance | | |
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他学科聴講の条件
Conditions of Other department student's attendance | | | This lecture is intended for master students from the graduate school of mathematics, but any motivated student can also attend it. |
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レベル
Level | | |
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キーワード
Keyword | | | Algebraic numbers, algebraic integer, algebraic number fields, Integral bases, Dedekind domains, Lattices & Minkowski Theory, Ideal class group, Dedekind unit theorem, Cyclotomic Fields, Fermat's Last Theorem |
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履修の際のアドバイス
Advice | | | If you are interested in number theory you should take this course. |
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授業開講形態等
Lecture format, etc. | | | 多-409 or online in Zoom. (Depending on the current situation). Please check the course homepage at the end of September for details. |
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遠隔授業（オンデマンド型）で行う場合の追加措置
Additional measures for remote class (on-demand class) | | |
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