This course is an exercise session for the lecture ``Introduction to Algebra I'' (ZUA.A201). The materials for exercise are chosen from that course.
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.
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
✔ Specialist skills | Intercultural skills | Communication skills | Critical thinking skills | Practical and/or problem-solving skills |
Students are given exercise problems related to what is taught in the course``Introduction to Algebra I'''.
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. |
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.
Shoichi Nakajima : Basics of Algebra and Arithmetic, Kyoritsu Shuppan Co., Ltd., 2000.
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.
Brief exam and presentation for exercise problems. Details will be announced during a lecture.
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.