TOKYO INSTITUTE OF TECHNOLOGY

Course description and aims

Algebra is a discipline of mathematics that deals with abstract notions which generalize algebraic operations on various mathematical objects. The main subjects 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.

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.

Student learning outcomes

To become familiar with important notions in algebra such as rings, subrings, fields, domains, ideals, residue rings, ring homomorphisms, the fundamental theorem on ring homomorphisms, the Chinese remainder theorem, Euclidean domains, principal 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, subribg, field, domain, ideal, residue ring, prime ideal, maximal ideal, ring homomorphism, the fundamental theorem on ring homomorphisms, the Chinese remainder theorem, Euclidean domain, principal ideal domain, prime elements and irreducible elements, unique factorization domain

Competencies that will be developed

Class flow

Standard lecture course

Course schedule/Required learning

Out-of-Class Study Time (Preparation and Review)

To enhance effective learning, students are encouraged to spend approximately 100 minutes preparing for class and another 100 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.

P.J. Cameron : Introduction to Algebra (second ed.), Oxford Univ. Press, 2008.
N. Jacobson : Basic Algebra I (second ed.), Dover，1985.
M. Artin : Algebra (second ed.), Addison-Wesley, 2011.
N. Herstein: Topics in algebra, John Wiley & Sons, 1975.
A. Weil: Number Theory for Beginners, Springer-Verlag, 1979.

Assessment criteria and methods

Midterm exam and final exam. Details will be announced during a lecture.

Related courses

• MTH.A201 ： Introduction to Algebra I
• MTH.A202 ： Introduction to Algebra II
• ZUA.A202 ： Exercises in Algebra A I
• ZUA.A203 ： Introduction to Algebra II
• ZUA.A204 ： Exercises in Algebra A II

Prerequisites (i.e., required knowledge, skills, courses, etc.)

Students are supposed to have completed [Linear Algebra I / Recitation], [Linear Algebra II] and [Linear Algebra Recitation II].
Students are strongly recommended to take ZUA.A202: Exercises in Algebra A I (if not passed yet) at the same time.

Other

Zoom URL for the online lecture classes is provided through the T2SCHOLA system.
Final exam will be in person.