Introduction to Algebra II

Academic unit or major

Undergraduate major in Mathematics

Instructor(s)

Satoshi Naito / Tatsuhiro Minagawa

Class Format

Lecture/Exercise (Face-to-face)

Media-enhanced courses

-

Day of week/Period
(Classrooms)

3-8 Fri

Class

-

Course Code

MTH.A202

Number of credits

110

Course offered

2025

Offered quarter

2Q

Syllabus updated

Mar 19, 2025

Language

Japanese

Syllabus

Course overview and goals

Algebra is a discipline of mathematics that deals with abstract notions which generalize algebraic operations on various mathematical objects. The main subjects of of this course include basic notions and properties of algebraic operations and of (commutative) rings, which are an abstraction/generalization of the integers and polynomials, and their ideals and residue rings. To help deeper understanding of the newly learnt concepts, each even-numbered class is devoted to a discussion session, where excercises are given related to the contents of the preceding lecture. This course succeeds "Introduction to Algebra I" offered in the first quater.

The contents of this course form not only a foundation of the whole Algebra but also an indispensable body of knowledge in other areas of mathematics such as Analysis and Geometry. Also, it is a basic attitude in all mathematical sciences to perform logical arguments without depending on intuition. In this course, we provide rigorous proofs, based on the notions of sets and maps, so that the students can learn how typical mathematical arguments should go.

Course description and aims

To become familiar with important notions in algebra such as ring homomorphisms, the fundamental theorem on ring homomorphisms, the Chinese remainder theorem, Euclidean domains, prime ideal domains, prime elements and irreducible elements, and unique factorization domains.

To become able to prove by him/herself basic properties of these objects.

Keywords

ring homomorphism, the fundamental theorem on ring homomorphims, Chinese remainder theorem, Euclidean domain, principal ideal domain, prime elements and irreducible elements, unique factorization domain

Competencies

• Specialist skills
• Intercultural skills
• Communication skills
• Critical thinking skills
• Practical and/or problem-solving skills

Class flow

Standard lecture course accompanied by discussion sesssions.

Course schedule/Objectives

Study advice (preparation and review)

To enhance effective learning, students are encouraged to spend approximately 100 minutes preparing for class and another 60 minutes reviewing class content afterwards (including assignments) for each class.
They should do so by referring to textbooks and other course material.

Textbook(s)

Shoichi Nakajima : Basics of Algebra and Arithmetic, Kyoritsu Shuppan, Co., Ltd., 2000.

Reference books, course materials, etc.

Akihiko Yukie: Algebra, No. 2; Rings, Fields, and the Galois Theory, Nippon Hyoron sha Co., Ltd., 2010.
M. Artin : Algebra (second ed.), Addison-Wesley, 2011.
A. Weil: Number Theory for Beginners, Springer-Verlag, 1979.

Evaluation methods and criteria

Based on evaluation of the results for discussion session and final examination. Details will be announced during a lecture.

Related courses

• MTH.A201 ： Introduction to Algebra I
• MTH.A203 ： Introduction to Algebra III
• MTH.A204 ： Introduction to Algebra IV
• MTH.A211 ： Advanced Linear Algebra I
• MTH.A212 ： Advanced Linear Algebra II

Prerequisites

Students are supposed to have completed [Linear Algebra I / Recitation], [Linear Algebra II], [Linear Algebra Recitation II] and [Introduction to Algebra I].

Contact information (e-mail and phone) Notice : Please replace from ”[at]” to ”@”(half-width character).

Naito: naito[at]math.titech.ac.jp
Minagawa: minagawa[at]math.titech.ac.jp