### 2020　Introduction to Algebra I

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Instructor(s)
Naito Satoshi  Somekawa Mutsuro  Minagawa Tatsuhiro
Course component(s)
Lecture / Exercise    (ZOOM)
Day/Period(Room No.)
Wed3-4(H112)  Thr5-8(H112)
Group
-
Course number
MTH.A201
Credits
2
2020
Offered quarter
1Q
Syllabus updated
2020/9/18
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 will be succeeded by ``Introduction to Algebra II'' in the second quarter.

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 such as the integer ring, polynomial rings, binary operations, equivalence relations, equivalence classes, residue rings of the integer ring, and residue rings of a polynomial ring.
To become able to prove by him/herself basic properties of these objects.

### Keywords

integer ring, polynomial ring, binary operation, equivalence relation, equivalence classe, residue rings of the integer ring, residue rings of a polynomial ring

### 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 Natural numbers, the integer ring, the rational number field, the real number field, the complex number field, polynomial rings Details will be provided during each class session.
Class 2 Discussion session on natural numbers, the integer ring, the rational number field, the real number field, the complex number field, polynomial rings Details will be provided during each class session.
Class 3 The integer ring, the residue theorem and factore theorem in a polynomial ring Details will be provided during each class session.
Class 4 Discussion session on the integer ring, the residue theorem and factore theorem in a polynomial ring Details will be provided during each class session.
Class 5 Basic notions of sets and maps, ordered pair, Cartesian product Details will be provided during each class session.
Class 6 Discussion session on basic notions of sets and maps, ordered pair, Cartesian product Details will be provided during each class session.
Class 7 Binary relations, binary operations Details will be provided during each class session.
Class 8 Discussion session on binary relations, binary operations Details will be provided during each class session.
Class 9 Equivalence relations, equivalence classes Details will be provided during each class session.
Class 10 Discussion session on equivalence relations, equivalence classes Details will be provided during each class session.
Class 11 Division of a set with respect to an equivalence relation Details will be provided during each class session.
Class 12 Discussion session on division of a set with respect to an equivalence relation Details will be provided during each class session.
Class 13 Residue rings of the integer ring, residue rings of a polynomial ring Details will be provided during each class session.
Class 14 Discussion session on residue rings of the integer ring, residue rings of a polynomial ring 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.A202 ： Introduction to Algebra II
• 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] and [Linear Algebra Recitation II].