### 2021　Exercises in Algebra A I

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Mathematics
Instructor(s)
Naito Satoshi  Minagawa Tatsuhiro  Wakabayashi Yasuhiro
Course component(s)
Exercise    (ZOOM)
Day/Period(Room No.)
Thr5-8(H112)  Fri5-8(H102)
Group
-
Course number
ZUA.A202
Credits
2
2021
Offered quarter
1-2Q
Syllabus updated
2021/3/19
Lecture notes updated
-
Language used
Japanese
Access Index

### Course description and aims

This course is an exercise session for the lecture ``Introduction to Algebra I'' (ZUA.A201). The materials for exercise are chosen from that course.

### 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, residue rings of a polynomial ring, the axiom of rings, subrings, ideals, residue rings, homomorphisms of rings, and the fundamental theorem on ring homomorphisms.

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, ring, subring, ideal, residue ring, homomorphism of rings, the fundamental theorem on ring homomorphims

### Competencies that will be developed

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

### Class flow

Students are given exercise problems related to what is taught in the course``Introduction to Algebra I'''.

### Course schedule/Required learning

Course schedule Required learning
Class 1 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 announced during each lecture.
Class 2 Discussion session on the integer ring, the residue theorem and factore theorem in a polynomial ring Details will be announced during each lecture.
Class 3 Discussion session on basic notions of sets and maps, ordered pair, Cartesian product Details will be announced during each lecture.
Class 4 Discussion session on binary relations, binary operations Details will be announced during each lecture.
Class 5 Discussion session on equivalence relations, equivalence classes Details will be announced during each lecture.
Class 6 Discussion session on division of a set with respect to an equivalence relation Details will be announced during each lecture.
Class 7 Discussion session on residue rings of the integer ring, residue rings of a polynomial ring Details will be announced during each lecture.
Class 8 evaluation of progress Details will be announced during each lecture.
Class 9 Discussion session on the axiom of rings, tyical examples of rings, and first properties of rings Details will be announced during each lecture.
Class 10 Discussion session on basic properties of the zero and inverse elements of a ring Details will be announced during each lecture.
Class 11 Discussion session on the definition of a subring, criterion for subrings, and examples of subrings Details will be announced during each lecture.
Class 12 Discussion session on homomorphisms of rings and their basic properties Details will be announced during each lecture.
Class 13 Discussion session on ideals of a ring Details will be announced during each lecture.
Class 14 Discussion session on residue rings and the fundamental theorem on ring homomorphisms Details will be announced during each lecture.

### 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

Brief exam and presentation for exercise problems. Details will be announced during a lecture.

### Related courses

• MTH.A201 ： Introduction to Algebra I
• MTH.A202 ： Introduction to Algebra II
• ZUA.A201 ： Introduction to Algebra 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.A201: Introduction to Algebra I (if not passed yet) at the same time.