Pozitif karakteristikteki fonksiyon cisimleri teorisine katkılar

In this thesis, we consider two problems related to the theory of function fi elds in positive characteristic. In the first part, we study the automorphisms of a function field of genus g at least 2 over an algebraically closed fi eld of characteristic p > 0. We show that for any nilpotent subgroup G of the automorphism group, the order of G is bounded by 16(g-1) when G is not a p-group and by 4pg^2/(p - 1)^2 when G is a p-group. Also, there are examples of function fields attaining these bounds; therefore, the bounds we obtained cannot be improved. In the second part, we focus on maximal function fields over finite fields having large automorphism groups. More precisely, we consider maximal function fields over the finite field Fp4 whose automorphism groups have order exceeding the Hurwitz's bound. We determine some conditions under which the maximal function eld is Galois covered by the Hermitian function field.