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.
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.
integer ring, polynomial ring, binary operation, equivalence relation, equivalence classe, residue rings of the integer ring, residue rings of a polynomial ring
|✔ Specialist skills||Intercultural skills||Communication skills||Critical thinking skills||Practical and/or problem-solving skills|
Standard lecture course accompanied by discussion sesssions.
|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|
|Class 3||The integer ring, the residue theorem and factore theorem in a polynomial ring|
|Class 4||Discussion session on the integer ring, the residue theorem and factore theorem in a polynomial ring|
|Class 5||Basic notions of sets and maps, ordered pair, Cartesian product|
|Class 6||Discussion session on basic notions of sets and maps, ordered pair, Cartesian product|
|Class 7||Binary relations, binary operations|
|Class 8||Discussion session on binary relations, binary operations|
|Class 9||Equivalence relations, equivalence classes|
|Class 10||Discussion session on equivalence relations, equivalence classes|
|Class 11||Division of a set with respect to an equivalence relation|
|Class 12||Discussion session on division of a set with respect to an equivalence relation|
|Class 13||Residue rings of the integer ring, residue rings of a polynomial ring|
|Class 14||Discussion session on residue rings of the integer ring, residue rings of a polynomial ring|
|Class 15||Checking session|
None in particular
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.
Based on evaluation of the results for discussion session and final examination. 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].