# Finite Fields (MATH332) Course Detail

Course Name Course Code Season Lecture Hours Application Hours Lab Hours Credit ECTS
Finite Fields MATH332 Area Elective 3 0 0 3 6
Pre-requisite Course(s)
Math 331
Course Language English Elective Courses Bachelor’s Degree (First Cycle) Face To Face Lecture, Question and Answer. This course is designed to introduce the basic theory of field extensions and finite fields which have various applications in both cryptography and coding theory. The students who succeeded in this course; gain knowledge about field extensions, be able to compute in a finite field, gain knowledge about polynomials over finite fields understand and use traces and norms in finite fields, learn how to factorize polynomials over finite fields. Characterization of finite fields, roots of irreducible polynomials, trace, norm, roots of unity and cyclotomic polynomials, order of polynomials and primitive polynomials, irreducible polynomials, construction of irreducible polynomials, factorization of polynomials

### Weekly Subjects and Releated Preparation Studies

Week Subjects Preparation
1 Algebraic Foundations: Groups, Rings (review) pp. 1-11
2 Algebraic Foundations: Fields, Polynomials pp. 11-30
3 Field Extensions pp. 30-37
4 Characterization of Finite Fields, Roots of Irreducible Polynomials pp. 45-51
5 Traces, Norms and Bases pp. 51-59
6 Roots of Unity and Cyclotomic Polynomials, Representation of Elements of Finite Fields pp. 60-66
7 Midterm Exam
8 Order of Polynomials and Primitive Polynomials pp. 76-84
9 Irreducible Polynomials, Construction of Irreducible Polynomials pp. 84-95
10 Examples of determining minimal polynomials pp. 96-100
11 Factorization over Small Finite Fields pp. 132-142
12 Factorization over Small Finite Fields (continued) pp. 132-142
13 Factorization over Large Finite Fields pp. 142-153
14 Calculation of Roots of Polynomials pp. 153-162
15 Review
16 Final Exam

### Sources

Course Book 1. Introduction to Finite Fields and their Applications, R. Lidl and H. Niederreiter, Cambridge University Press, 1994. 2. Applications of Finite Fields , Alfred J. Menezes, Ian F. Blake, Xuhong Gao, Ronald C. Mullin, Kluwer, 1993.

### Evaluation System

Attendance/Participation - -
Laboratory - -
Application - -
Field Work - -
Special Course Internship - -
Quizzes/Studio Critics - -
Homework Assignments 5 10
Presentation - -
Project - -
Report - -
Seminar - -
Midterms Exams/Midterms Jury 2 50
Final Exam/Final Jury 1 40
Toplam 8 100
 Percentage of Semester Work 60 40 100

### 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 16 3 48
Presentation/Seminar Prepration
Project
Report
Homework Assignments 5 8 40
Quizzes/Studio Critics
Prepration of Midterm Exams/Midterm Jury 2 12 24
Prepration of Final Exams/Final Jury 1 18 18