Additive polynomials are one of the main objects in function field arithmetic. When the field k of coefficients is infinite, additive polynomials can be viewed as elements in the skew polynomial ring k, where tau is the Frobenius morphism. In this talk, I will discuss several analogies between additive polynomials over function fields, and polynomials over number fields. In particular, I will focus on function field analogues of Bang-Zsigmondy’s theorem, Feit’s theorem, and Grunwald-Wang’s theorem when considering the simplest additive polynomial, T+tau, known as the Carlitz module, in the skew polynomial ring F_q(T), where F_q is the finite field with q elements, and T is an indeterminate.