Introduction to Number Theory(MAT364)
| Course Code | Course Name | Semester | Theory | Practice | Lab | Credit | ECTS |
|---|---|---|---|---|---|---|---|
| MAT364 | Introduction to Number Theory | 6 | 3 | 0 | 0 | 3 | 5 |
| Prerequisites | |
| Admission Requirements |
| Language of Instruction | English |
| Course Type | Elective |
| Course Level | Bachelor Degree |
| Course Instructor(s) | Ayberk ZEYTİN azeytin@gsu.edu.tr (Email) |
| Assistant | |
| Objective | In this course, the ultimate goal is to prove the quadratic reciprocity law using characters and Gauss sums. |
| Content | The course covers the fundamental concepts and key tools of the field, linking theory to applications through guided examples and progressively challenging exercises. |
| Course Learning Outcomes | By the end of the course, students will be able to understand and formalize the essential concepts, solve typical problems independently, and apply the methods studied to concrete situations related to the quadratic reciprocity law. |
| Teaching and Learning Methods | Teaching alternates between structured lectures, in-class problem solving, discussions, and practical work. Regular assignments and targeted feedback help consolidate learning. |
| References |
William Stein, Elementary Number Theory: Primes, Congruences, and Secrets, https://wstein.org/ent/ent.pdf Kenneth Ireland & Michael Rosen, A Classical Introduction to Modern Number Theory Ivan Niven, Herbert Zuckerman, Hugh Montgomery, An Introduction to the Theory of Numbers |
Theory Topics
| Week | Weekly Contents |
|---|---|
| 1 | Congruences; arithmetic mod $n$; guiding examples and conjectures. |
| 2 | Euclid's algorithm and B\'ezout's identity. GCD; modular inverses; linear congruences. |
| 3 | Prime numbers and unique factorization. Basic lemmas; applications to congruences. |
| 4 | The group $(\mathbb{Z}/n\mathbb{Z})^\times$. Euler's $\varphi$ function; Euler's theorem; order of an element. |
| 5 | Quadratic residues: exploration. Squares modulo a prime; counting; first residue tables. |
| 6 | Legendre symbol and Euler's criterion. Definition; multiplicativity; fast computations. |
| 7 | Supplementary laws. $\left(\frac{-1}{p}\right)$ (mod $4$) and $\left(\frac{2}{p}\right)$ (mod $8$); guided proofs. |
| 8 | Multiplicative characters. Characters of $(\mathbb{Z}/p\mathbb{Z})^\times$; orthogonality; the quadratic character. |
| 9 | Additive characters and roots of unity. Exponential sums modulo $p$; basic identities. |
| 10 | Gauss sums I. Definition $\tau(\chi)$; twisting identities; absolute value and examples. |
| 11 | Gauss sums II. Evaluation of the quadratic Gauss sum; determining the sign via $p \bmod 4$. |
| 12 | Quadratic reciprocity. Proof via Gauss sums and characters; synthesis of the steps. |
| 13 | Jacobi symbol and effective computation. Generalization to composite denominators; caveats and examples. |
| 14 | Applications & projects. Deciding solvability of $x^2\equiv a \pmod p$; mini-projects and final portfolios. |
Practice Topics
| Week | Weekly Contents |
|---|
Contribution to Overall Grade
| Number | Contribution | |
|---|---|---|
| Contribution of in-term studies to overall grade | 6 | 60 |
| Contribution of final exam to overall grade | 1 | 40 |
| Toplam | 7 | 100 |
In-Term Studies
| Number | Contribution | |
|---|---|---|
| Assignments | 6 | 60 |
| Presentation | 0 | 0 |
| Midterm Examinations (including preparation) | 0 | 0 |
| Project | 0 | 0 |
| Laboratory | 0 | 0 |
| Other Applications | 0 | 0 |
| Quiz | 0 | 0 |
| Term Paper/ Project | 0 | 0 |
| Portfolio Study | 0 | 0 |
| Reports | 0 | 0 |
| Learning Diary | 0 | 0 |
| Thesis/ Project | 0 | 0 |
| Seminar | 0 | 0 |
| Other | 0 | 0 |
| Make-up | 0 | 0 |
| Toplam | 6 | 60 |
| No | Program Learning Outcomes | Contribution | ||||
|---|---|---|---|---|---|---|
| 1 | 2 | 3 | 4 | 5 | ||
| 1 | understands principles of deductive reasoning; has experience to verify well-foundedness and exactness of mathematical statements in systematic ways; | X | ||||
| 2 | can properly state and use concepts and results of major mathematical interest; | X | ||||
| 3 | masters current computational techniques and algorithms; has a good ability in their use; can identify relevant tools, among those one has learned, suitable to solve a problem and is able to judge whether or not one is in possession of these tools; | X | ||||
| 4 | is able to express one’s mathematical ideas in an organised way both in written and oral forms; | X | ||||
| 5 | understands relations connecting substantial concepts and results; can switch from one viewpoint to another on mathematical objects (pictures, formulae, precise statements, heuristic trials, list of examples,...); | X | ||||
| 6 | has followed individually a guided learning strategy; has pursued steps toward the resolution of unfamiliar problems; | X | ||||
| 7 | has a theoretical and practical knowledge in computer science well adapted for learning a programming language; | X | ||||
| 8 | has investigated the relevance of modeling and using mathematical tools in natural sciences and in the professional life; is conscious about historical development of mathematical notions; | X | ||||
| 9 | has followed introduction to some mathematical or non-mathematical disciplines after one’s proper choice; had experience to learn selected subjects according to one’s proper arrangement; | X | ||||
| 10 | masters French language as well as other foreign languages, to a level sufficient to study or work abroad. | X | ||||
| Activities | Number | Period | Total Workload |
|---|---|---|---|
| Class Hours | 14 | 3 | 42 |
| Working Hours out of Class | 14 | 2 | 28 |
| Assignments | 6 | 6 | 36 |
| Presentation | 0 | 0 | 0 |
| Midterm Examinations (including preparation) | 0 | 0 | 0 |
| Project | 0 | 0 | 0 |
| Laboratory | 0 | 0 | 0 |
| Other Applications | 0 | 0 | 0 |
| Final Examinations (including preparation) | 1 | 20 | 20 |
| Quiz | 0 | 0 | 0 |
| Term Paper/ Project | 0 | 0 | 0 |
| Portfolio Study | 0 | 0 | 0 |
| Reports | 0 | 0 | 0 |
| Learning Diary | 0 | 0 | 0 |
| Thesis/ Project | 0 | 0 | 0 |
| Seminar | 0 | 0 | 0 |
| Other | 0 | 0 | 0 |
| Make-up | 0 | 0 | 0 |
| Yıl Sonu | 0 | 0 | 0 |
| Hazırlık Yıl Sonu | 0 | 0 | 0 |
| Hazırlık Bütünleme | 0 | 0 | 0 |
| Total Workload | 126 | ||
| Total Workload / 25 | 5.04 | ||
| Credits ECTS | 5 | ||