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Identifier

000344160

Title

Το θεώρημα των πρώτων αριθμών σε σώματα συναρτήσεων

Alternative Title

The Prime Number Theorem

Author

Καπετανάκης, Γεώργιος Νικολάου

Thesis advisor

Γαρεφαλάκης, Θεόδουλος

Reviewer

Λυδάκης Μανώλης
Τζανάκης Νίκος

Abstract

Classic Number Theory is studying the set of integers Z. In order to solve several problems of classic Number Theory, Algebraic Number Theory was evolved and Algebraic Number Theory is mainly studying Z’s field of fractions, Q, the set of rational numbers and it’s algebraic finite extensions, the number fields. Inspired from the similarities between the sets Z and F[x], the set of polunomials of one variable over a finite field F, we are going to imitate the above constructions. So we are going to study the field F(x), the field of rational functions over a finite field F, and its algebraic finite extensions, the function fields. We will show the analogues of several theorems of classic Number Theory, like the Prime Number Theorem, or even the analogues of open problems, like the Riemann Hypothesis. The above constructions can be studied using Algebraic Geometry, or even Complex Analysis, but we keep a more number theoretic aspect of the subject. In several cases many results apply even if we substitute F with the arbitary field. We will try not to limit the power of such results, if possible.

Language

Greek

Subject

Algebra

Finite Fields

Function Fields

Number Theory

Άλγεβρα

Θεωρία Αριθμών

Θεώρημα Hasse-Weil

Θεώρημα Riemann-Roch Υπόθεση Riemann

Πεπερασμένα Σώματα