Introduction I.Equations yd=f(x)and yq-y=f(x) 1.Finite Fields 2.Equations yd=t(x) 3.Construction of certain polynomiale 4.Proof of the Main Theorem 5.Removal of the condition (m,d)=1 6.Hyperderivatives 7.Removal of the condition that q=p or p2 . 9.Equations yq-y=f(x) II.Character Sums and Exponential Sums 1.Characters of Finite Abelian Groups 2.Characters and Character Sums associated with Finite Flelds 3.Gaussian sums 4.The low road. 5.Systems of equations y1d1=f1(x), ,yndn=fn��x��. 6.Auxiliary lemmas on wv1+��+�� 7.Further auxillary lemmas 8.Zeta Function and L-Functions 9.special L-Functions 10.Fleld extengiong.The Davenport-Hasse relations 11.Proof of the Principal Theorems 12.Kloosterman Sums 13.Further Results III.Abeolutoly Irreducible Equatione f(x,y)=0 1.Introduction 2.Independence results 3.Derivatives. 4.Construction of two algebraic functions 5.Constructlon of two polynomlals 6.Proof of the Main Theorem 8.Hyperderivatives again 9.Removal of the condition that q=p IV.Equations in Many Variables 1.Theorems of Chevalley and Warning 2.Quadratic forms 3.Elementary upper bounds.Projective zeros 4.The average number of zeros of a polynomial 5.Additive Bquatione:A Chebychev Argument 6.Additive Equations:Character Sums 7.Bquations f,(y��x,l1+ +f(y��xn=o V.Absolutely Irreducible squations f(x,.��,x)=0 1.Elimination Theory 2.The absolute irreducibility of polynomlals(I) 3.The abaolute irreducibllity of polynomial8(II) 4.The absolute 1rreducibility of polynomlals(III) 5.The number of zeroe of abaolutely irreducib1e polynomiale in n varlables VI.Rudimente of Algebraic Goometry,The Number of Pointa in Varieties over Finite Fields 1.Varietoes 2.Dimension 3.Rational Maps 4.BirationalMaps 5.Linear Disjointness of Fields 6.Constant Field Extensions 7.Counting Points in Varieties Over Finite Pields