In this thesis, we consider the proof of an analogue of the Riemann hypothesis for a function field F of one 

variable with a finite constant field K. The classical Riemann Hypothesis ( which is still unproven ) says that 

-besides the trivial zeros  s = -2 ,-4 ,-6,...- all zeros of the Riemann-zeta function z(s) lie on the line  

Re(s)=1/2. In the function field case it was proved that the zeros of the function z _{F / K} (s) lie on the line  

Re(s)=1/2. Our aim in this thesis is to give the proof of this result, known as the Hasse-Weil Theorem, 

following the proof given by Bombieri.F / K 

Key words: Function Fields, Zeta Function, Riemann Hypothesis