### 2016　Exercises in Algebra A I

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Mathematics
Instructor(s)
Taguchi Yuichiro  Kawachi Takeshi  Minagawa Tatsuhiro
Course component(s)
Exercise
Day/Period(Room No.)
Thr5-8(H112)
Group
-
Course number
ZUA.A202
Credits
2
2016
Offered quarter
1-2Q
Syllabus updated
2016/4/27
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
Class 3 Discussion session on basic notions of sets and maps, ordered pair, Cartesian product
Class 4 Discussion session on binary relations, binary operations
Class 5 Discussion session on equivalence relations, equivalence classes
Class 6 Discussion session on division of a set with respect to an equivalence relation
Class 7 Discussion session on residue rings of the integer ring, residue rings of a polynomial ring
Class 8 evaluation of progress
Class 9 Discussion session on the axiom of rings, tyical examples of rings, and first properties of rings
Class 10 Discussion session on basic properties of the zero and inverse elements of a ring
Class 11 Discussion session on the definition of a subring, criterion for subrings, and examples of subrings
Class 12 Discussion session on homomorphisms of rings and their basic properties
Class 13 Discussion session on ideals of a ring
Class 14 Discussion session on residue rings and the first fundamental theorem on ring homomorphisms
Class 15 Discussion session on the second and third fundamental theorems on ring homomorphismsChecking session

### Textbook(s)

None in particular

### 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 oral 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.