| Course title | |||||
| 代数学Ⅰ [Algebra Ⅰ] | |||||
| Course category | technology speciality courses,ets. | Requirement | Credit | 2 | |
| Department | Year | 2～4 | Semester | Fall | |
| Course type | Fall | Course code | 022613 | ||
| Instructor(s) | |||||
| 與口 卓志 [YOGUCHI Takashi] | |||||
| Facility affiliation | Graduate School of Agriculture | Office | Email address | ||
| Course description |
| In modern engineering, the methods of abstract algebra based on sets and axioms are frequently used for cryptology, coding theory, formal language theory, etc. In this course, we will learn fundamentals of abstract algebra such as groups, rings and fields. |
| Expected Learning |
|
The goal of this course is to be capable of: (1) explaining the meanings of basic concepts and terms about groups, rings and fields, (2) performing basic operations on integers modulo n, (3) solving systems of linear congruence equations, (4) applying Fermat’s little theorem and Euler’s theorem to concrete examples. |
| Course schedule |
|
1. Sets and maps, the direct product of sets (pp.8-16) 2. Surjections and injections, the definition of algebraic systems (pp.15-18, p.22) 3. Equivalent relations and quotient sets (pp.19-20) 4. Semigroups and groups, examples of groups (pp.23-27, pp.44-48) 5. Subgroups, cyclic groups, equivalent relations induced by subgroups (pp.27-29, p.42) 6. Normal subgroups and quotient groups (p.28, pp.40-42 except for homomorphisms) 7. Homomorphisms and isomorphisms of groups, the fundamental theorem on homomorphisms (pp.36-42) 8. Exercises I 9. Rings and fields (pp.30-34) 10. Ideals of rings and quotient rings (pp.50-54) 11. Euclidean algorism and the inverse of an integer modulo n (pp.69-74) 12. Systems of linear congruence equations, Chinese remainder theorem (pp.74-76, p.149) 13. Fermat's little theorem and Euler's theorem, Fermat primality test (pp.86-92) 14. Exercises II 15. Term examination |
| Prerequisites |
| Nothing in particular |
| Required Text(s) and Materials |
| Mariko Hagita, “Angou no Tame no Daisuu Nyuumon”, Saiensu-sha (in Japanese) |
| References |
|
Ryuuichi Hirabayashi, “Kougaku Kiso Daisuukei to Sono Ouyou”, Suurikougaku-sha (in Japanese) Naoki Kawata, “Seisuu to Gun, Kan, Tai”, Gendai-Suugakusha (in Japanese) Koukichi Sugiwara, Toshiyuki Imai, “Kougaku no Tame no Ouyou Daisuu”, Kyouritsu Shuppan (in Japanese) |
| Assessment/Grading |
|
The term examination will constitute 80% of your grade and two exercises (the 8th and 14th weeks) constitute 20% of your grade. (Grade distribution of last year: S 4% A 25% B 20% C 27% D 24%) |
| Message from instructor(s) |
| For learners, algebra might seem too abstract at first. However, it is closely linked to well-known integers or polynomials. I will make efforts to give concrete examples in the lecture. |
| Course keywords |
| Algebraic system, Group, Ring, Field, Homomorphism, Normal subgroup, Ideal, Equivalent class, Congruence equation |
| Office hours |
| Remarks 1 |
| Remarks 2 |
| Related URL |
| Lecture Language |
| Language Subject |
| Last update |
| 9/19/2018 8:58:58 AM |