Algebra is a discipline of mathematics that deals with abstract notions which generalize algebraic operations on various mathematical objects. The main subjects 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.
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.
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
Intercultural skills | Communication skills | Specialist skills | Critical thinking skills | Practical and/or problem-solving skills |
---|---|---|---|---|
- | - | ✔ | - | - |
Standard lecture course
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 announced during each lecture. |
Class 2 | The integer ring, the residue theorem and factore theorem in a polynomial ring | Details will be announced during each lecture. |
Class 3 | Basic notions of sets and maps, ordered pair, Cartesian product | Details will be announced during each lecture. |
Class 4 | Binary relations, binary operations | Details will be announced during each lecture. |
Class 5 | Equivalence relations, equivalence classes | Details will be announced during each lecture. |
Class 6 | Division of a set with respect to an equivalence relation | Details will be announced during each lecture. |
Class 7 | 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 | Axiom of rings, tyical examples of rings, and first properties of rings | Details will be announced during each lecture. |
Class 10 | Basic properties of the zero and inverse elements of a ring | Details will be announced during each lecture. |
Class 11 | Definition of a subring, criterion for subrings, and examples of subrings | Details will be announced during each lecture. |
Class 12 | Homomorphisms of rings and their basic properties | Details will be announced during each lecture. |
Class 13 | Ideals of a ring | Details will be announced during each lecture. |
Class 14 | Residue rings and the first fundamental theorem on ring homomorphisms | Details will be announced during each lecture. |
Class 15 | The second and third fundamental theorems on ring homomorphisms | Details will be announced during each lecture. |
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.
Midterm exam and final exam. 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.A202: Exercises in Algebra A I (if not passed yet) at the same time.