MATH503 ALGEBRA I
| Course Code: | 2360503 |
| METU Credit (Theoretical-Laboratory hours/week): | 3 (3.00 - 0.00) |
| ECTS Credit: | 8.0 |
| Department: | Mathematics |
| Language of Instruction: | English |
| Level of Study: | Graduate |
| Course Coordinator: | Prof.Dr. ÖMER KÜÇÜKSAKALLI |
| Offered Semester: | Fall Semesters. |
Course Objectives
The objective of this course is to develop a rigorous understanding of fundamental structures in abstract algebra, including groups, rings, and their associated concepts. Students will explore advanced topics such as quotient groups, group actions, Sylow theorems, solvable and nilpotent groups, as well as key ring-theoretic concepts including ideals, homomorphisms, factorization, and localization. Emphasis will be placed on understanding the structure and classification of algebraic systems through the use of isomorphism theorems, direct products, and generators and relations. The course also aims to equip students with the ability to apply theoretical concepts to polynomial rings, principal ideal domains, and unique factorization domains, laying a solid foundation for further study in pure or applied algebra.
Course Content
Groups quotient groups, isomorphism theorems, alternating and dihedral groups, direct products, free groups, generators and relations, free abelian groups, actions. Sylow theorems, nilpotent and solvable groups, normal and subnormal series. Rings, ring homomorphisms, ideals, factorization in commutative rings, rings of quotients, localization, principle ideal domains, Eucladian domains, unique factorization domains, polynomials and formal power series, factorization in polynomial rings.
Course Learning Outcomes
By the end of this course, students will be able to:
- Recognize and analyze algebraic structures in both theoretical and applied contexts.
- Communicate mathematical reasoning and formal definitions clearly and effectively.
- Define and analyze group structures, including subgroups, normal subgroups, and quotient groups.
- Apply and prove the Isomorphism Theorems for groups.
- Explain and apply group actions and the Orbit-Stabilizer Theorem.
- Use the Sylow theorems to classify finite groups and analyze their subgroup structures.
- Understand the structure and properties of rings, subrings, ring homomorphisms, and ideals.
- Construct and analyze quotient rings and perform localization.
- Explain and identify properties of Euclidean domains, principal ideal domains (PIDs), and unique factorization domains (UFDs).
Program Outcomes Matrix
| Level of Contribution | |||||
| # | Program Outcomes | 0 | 1 | 2 | 3 |
| 1 | Acquires mathematical thinking skills (problem solving, generating ways of thinking, forming correspondence, generalizing etc.) and can use them in related fields. | ✔ | |||
| 2 | Gains academic maturity through self-study. | ✔ | |||
| 3 | Can design mathematics related problems, devise solution methods and apply them when appropriate. | ✔ | |||
| 4 | Carries out parts of a mathematical research program independently. | ✔ | |||
| 5 | Has a command of Turkish and English languages so that he/she can actively communicate (read, write, listen and speak). | ✔ | |||
| 6 | Contributes to solving global, environmental and social problems either individually or as being part of a social group. | ✔ | |||
| 7 | Respects ethical values and rules; applies them in professional and social issues. | ✔ | |||
| 8 | Can work cooperatively in a team and also individually. | ✔ | |||
| 9 | Gets exposed to academic culture through interaction with others. | ✔ | |||
| 10 | Comprehends necessity of knowledge, can define it and acquires it; uses knowledge effectively and shares it with others. | ✔ | |||
0: No Contribution 1: Little Contribution 2: Partial Contribution 3: Full Contribution