It is proved that certain polynomials of degree ≱3 have a cyclic group of order n as Galois group over all rational function fields k(t) with characteristic not dividing n. Moreover, the extension fields of k(t) generated by the polynomials have k as precise field of constants, and possess an unramified rational point. For all 3≤≰20 with the exceptions of 17 and 19 the polynomials are calculated explicitly