# Abstract Algebra (MATH331) Course Detail

Course Name Course Code Season Lecture Hours Application Hours Lab Hours Credit ECTS
Abstract Algebra MATH331 5. Semester 4 0 0 4 7
Pre-requisite Course(s)
MATH 111 Basic Logic and Algebra
Course Language English Compulsory Departmental Courses Bachelor’s Degree (First Cycle) Face To Face Lecture, Question and Answer. The course is designed to provide necessary backgrounds in Abstract Algebra. In this course students will learn the general concepts of abstract algebra. The students who succeeded in this course; understand basics of group theory, understand basics of ring theory, understand basics of field theory. Groups: subgroups, cyclic groups, permutation groups, Lagrange Theorem, normal subgroups and factor groups, homomorphisms, isomorphism theorems, rings and fields: subrings, integral domains, ideals and factor rings, maximal and prime ideals, homomorphisms of rings,field of quotients, polynomial rings, principal ideal domain (PID), irreducibility of

### Weekly Subjects and Releated Preparation Studies

Week Subjects Preparation
1 Introduction to Groups (Symmetries of a Square, The Dihedral Groups: pg 31-37) Groups: Definition of group and Abelian group, basic examples (42-49) pp. 31-37, 42-49
2 Elementary properties of Groups (uniquness of identity, cancellation, uniquness of inverse element, inverse of the product: 50-53) Finite groups; Subgroups: Order of a Group, order of an element, definition of subgroup and basic examples (59-66) pp. 50-53, 59-66
3 Cyclic Groups (73-82) Permutation groups (94-112) pp. 73-82, 94-112
4 Isomorphisms: Definition and examples,Cayley’s Theorem, Properties of Isomorphism (120-128 skip automorphisms but the definition) Cosets and the Lagrange Theorem: Definition, Properties of Cosets, Lagrange Theorem (137-141 up to Fermat’s Little Theorem) pp. 120-128, 137-141
5 External Direct Product: Definitions and Examples, Properties of Direct Product (153-157), Applications* Normal subgroups and Factor Groups (177-184), Internal Direct Products (187-190) pp. 153-157, 177-184, 187-190
6 Group Homomorphisms: Definitions, Examples, Properties of Homomorphisms, The First Isomorphism Theorem (199-207) Fundamental Theorem of Finite Abelian Groups (217-225) pp. 199-207,217-225
7 Rings: Definition of Ring and examples, properties of ring, Uniqueness of Unity and Inverses, Definition of subring, Subring test (235-240) pp. 235-240
8 Integral Domains: Definition of zero-divisors and integral domain with examples, Cancellation Theorem, Definition of Field, Finite Integral Domains are Field, Z_p is a field, (248-251) Ideals and Factor Rings: Definition, Ideal Test, Existance of Factor Rings, Examples (261-265) pp. 248-251, 261-265
9 Prime Ideals and Maximal Ideals (266-268) Ring Homomorphisms: Definitions with examples, properties of ring homomorphisms, First Isomorohism Theorem for Rings (278-284) pp. 266-268, 278-284
10 Field of Quotients(284-285) Polynomial Rings (291-294) The Division Algorithm and Consequences: Division Algorithm for F[x], The Remainder Theorem, The Factor Theorem (294-297) pp. 284-285,291-294, 294-297
11 Principal Ideal Domain (PID), F[x] is a PID (297-298) pp. 297-298
12 Factorization of Polynomials: Definition of Irreducible and reducible polynomials, Reducibility test for degrees 2 and 3, Content of a Polynomial, Primitive Polynomial, Gauss Lemma (303-306) Irreducibility Test: Mod p irreducibility test, Eisenstein’s Criterion (306-311) pp. 303-306, 306-311
13 Divisibility in Integral Domains:Irreducibles, Primes (320-323), Unique Factorization Domains (326-329), pp. 326-329
14 Euclidean Domains (329-333) pp. 329-333

### Sources

Course Book 1. Contemporary Abstract Algebra, by Joseph A. Gallian 2. A First Course in Abstract Algebra, by John B. Fraleigh 3. Fundamentals of Abstract Algebra, by D.S. Malik, John M. Morderson, M.K. Sen , McGraw-Hill

### Evaluation System

Attendance/Participation - -
Laboratory - -
Application - -
Field Work - -
Special Course Internship - -
Quizzes/Studio Critics - -
Homework Assignments 4 10
Presentation - -
Project - -
Report - -
Seminar - -
Midterms Exams/Midterms Jury 2 55
Final Exam/Final Jury 1 35
Toplam 7 100
 Percentage of Semester Work 65 35 100

### Course Category

Core Courses X

### The Relation Between Course Learning Competencies and Program Qualifications

# Program Qualifications / Competencies Level of Contribution
1 2 3 4 5
1 Has the ability to apply scientific knowledge gained in the undergraduate education and to expand and extend knowledge in the same or in a different area X
2 Can apply gained knowledge and problem solving abilities in inter-disciplinary research X
3 Has the ability to work independently within research area, to state the problem, to develop solution techniques, to solve the problem, to evaluate the obtained results and to apply them when necessary X
4 Takes responsibility individually and as a team member to improve systematic approaches to produce solutions in unexpected complicated situations related to the area of study X
5 Can develop strategies, implement plans and principles on the area of study and can evaluate obtained results within the framework X
6 Can develop and extend the knowledge in the area and to use them with scientific, social and ethical responsibility X
7 Has the ability to follow recent developments within the area of research, to support research with scientific arguments and data, to communicate the information on the area of expertise in a systematically by means of written report and oral/visual presentation X
8 To have an oral and written communication ability in at least one of the common foreign languages ("European Language Portfolio Global Scale", Level B2) X
9 Has software and hardware knowledge in the area of expertise, and has proficient information and communication technology knowledge X
10 Follows scientific, cultural, and ethical criteria in collecting, interpreting and announcing data in the research area and has the ability to teach. X
11 Has professional ethical consciousness and responsibility which takes into account the universal and social dimensions in the process of data collection, interpretation, implementation and declaration of results in mathematics and its applications. X

Activities Number Duration (Hours) Total Workload
Course Hours (Including Exam Week: 16 x Total Hours)
Laboratory
Application
Special Course Internship
Field Work
Study Hours Out of Class 14 4 56
Presentation/Seminar Prepration
Project
Report
Homework Assignments 4 8 32
Quizzes/Studio Critics
Prepration of Midterm Exams/Midterm Jury 2 16 32
Prepration of Final Exams/Final Jury 1 22 22