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|
|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|
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.
Brief exam and oral 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.