Thesis (MSc)--Stellenbosch University, 2011. / ENGLISH ABSTRACT: It is well known that the ring of integers Z and the ring of polynomials A = Fr[T] over a finite field Fr have many properties in common. It is due to these properties that almost all the famous (multiplicative) number theoretic results over Z have analogues over A. In this thesis, we are devoted to utilising this analogy together with the theory of Carlitz modules. We do this to survey and compare the analogues of cyclotomic polynomials, the size of their coefficients and cyclotomic extensions over the rational function field k = Fr(T). / AFRIKAANSE OPSOMMING: Dit is bekend dat Z, die ring van heelgetalle en A = Fr[T], die ring van polinome oor ’n eindige liggaam baie eienskappe in gemeen het. Dit is as gevolg van hierdie eienskappe dat feitlik al die bekende multiplikative resultate wat vir Z geld, analoë in A het. In hierdie tesis, fokus ons op die gebruik van hierdie analogie saam met die teorie van die Carlitz module. Ons doen dit om ’n oorsig oor die analoë van die siklotomiese polinome, hul koëffisiënte, en siklotomiese uitbreidings oor die rasionele funksie veld k = Fr(T).