This thesis explores arithmetic aspects of function fields, viewed as analogues of number fields. We begin by reviewing the classical theory: valuation rings, places, and divisors, concluding with the Riemann–Roch theorem and its consequences. We then study algebraic extensions of function fields, including Galois extensions, the Hurwitz genus formula, and constant field extensions, with an application to elliptic function fields. A second part develops the theory of characters, leading to applications of Gauss and Jacobi sums in solving Diophantine equations over finite fields. The main contribution is presented in the final chapter: a function field analogue of results on the divisor function and B-free integers. We introduce B-free polynomials and establish asymptotic estimates for averages of the divisor function, extending results of Camargo [dC24] to the function field setting.