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2022　Introduction to Algebra II

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Academic unit or major
Undergraduate major in Mathematics
Instructor(s)
Naito Satoshi  Suzuki Masatoshi  Minagawa Tatsuhiro 
Class Format
Lecture / Exercise    (Blended)
Media-enhanced courses
Day/Period(Room No.)
Fri3-4(H103)  Fri5-8(H103)  
Group
-
Course number
MTH.A202
Credits
2
Academic year
2022
Offered quarter
2Q
Syllabus updated
2022/3/16
Lecture notes updated
-
Language used
Japanese
Access Index

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 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.

Student learning outcomes

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 that will be developed

✔ 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/Required learning

  Course schedule Required learning
Class 1 Definition and typical examples of ring homomorphisms Details will be provided during each class session.
Class 2 Discussion session on the definition and typical examples of ring homomorphisms Details will be provided during each class session.
Class 3 First fundamental theorem on ring homomorphisms and its applications Details will be provided during each class session.
Class 4 Discussion session on the first fundamental theorem on ring homomorphisms and its applications Details will be provided during each class session.
Class 5 Chinese remainder theorem and its applications Details will be provided during each class session.
Class 6 Discussion session on the Chinese remainder theorem and its applications Details will be provided during each class session.
Class 7 Definition and typical rxamples of Euclidean domains Details will be provided during each class session.
Class 8 Discussion session on the definition and examples of Euclidean domains Details will be provided during each class session.
Class 9 Definition and typical examples of principal ideal domains Details will be provided during each class session.
Class 10 Discussion session on the definition and typical examples of principal ideal domains Details will be provided during each class session.
Class 11 Definitions and basic properties of prime elements and irreducible elements Details will be provided during each class session.
Class 12 Discussion session on the definition and basic properties of prime elements and irreducible elements Details will be provided during each class session.
Class 13 Unique factorization domains Details will be provided during each class session.
Class 14 Discussion session on unique factorization domains Details will be provided during each class session.

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

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

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

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

Other

Zoom URL for the online lecture classes is provided throught the T2SCHOLA system.
Final exam will be in person.
Also, the recitation classes will be in person.

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