Algebraic proofs of Dirichlet's theorem on arithmetic progressions
| Thesis title in Czech: | Algebraické důkazy Dirichletovy věty o aritmetických posloupnostech |
|---|---|
| Thesis title in English: | Algebraic proofs of Dirichlet's theorem on arithmetic progressions |
| Key words: | Dirichletova věta, algebraická teorie čísel, prvočíslo, Chebotarevova věta o hustotě |
| English key words: | Dirichlet's theorem, algebraic number theory, primes, Chebotarev Density Theorem |
| Academic year of topic announcement: | 2015/2016 |
| Thesis type: | Bachelor's thesis |
| Thesis language: | angličtina |
| Department: | Department of Algebra (32-KA) |
| Supervisor: | doc. Mgr. Vítězslav Kala, Ph.D. |
| Author: | hidden - assigned and confirmed by the Study Dept. |
| Date of registration: | 13.10.2015 |
| Date of assignment: | 15.10.2015 |
| Confirmed by Study dept. on: | 07.12.2015 |
| Date and time of defence: | 17.06.2016 00:00 |
| Date of electronic submission: | 26.05.2016 |
| Date of submission of printed version: | 27.05.2016 |
| Date of proceeded defence: | 17.06.2016 |
| Opponents: | doc. Mgr. Pavel Příhoda, Ph.D. |
| Guidelines |
| Dirichlet's theorem says that there are infinitely many primes in every arithmetic progression ax+b with coprime a and b. The general proof is analytic, but in certain special cases it is possible to give more elementary, algebraic proofs. The goal of the thesis is to work out in detail the Euclidean proofs approach of [1] and solve related exercises. Besides from learning the basics of algebraic number theory this also requires non-trivial knowledge of Galois theory.
The student may then possibly continue to study the generalization of this approach to arithmetic progressions in other number fields (involving applications of Tchebotarev density theorem). |
| References |
| [1] M. R. Murty, N. Thain. Prime numbers in certain arithmetic progressions: Funct. Approx. Comment. Math. 35 (2006), 249–259.
[2] K. Conrad: Euclidean proofs of Dirichlet's theorem, www.math.uconn.edu/~kconrad/blurbs/dirichleteuclid.pdf. [3] J. S. Milne: Algebraic Number Theory, http://www.jmilne.org/math/CourseNotes/ant.html. [4] M. R. Murty, J. Esmonde: Problems in Algebraic Number Theory, GTM 190. [5] S. Lang: Algebraic Number Theory, GTM 110. |