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One of the main themes in this thesis is the description of the signature ofboth the infinite place and the finite places in cubic function fields of anycharacteristic and quartic function fields of characteristic at least 5. Forthese purposes, we provide a new theory which can be applied to cubic andquartic function fields and to even higher dimensional function fields. One ofthe striking advantages of this theory to other existing methods is that isdoes not use the concept of p-adic completions and we can dispense of Cardano'sformulae. Another key result comprises the construction of cubic functionfields of unit rank 1 and 2, with an obvious fundamental system. One of themain ingredients for such constructions is the definition of the maximum value.This definition is new and very prolific in the context of finding fundamentalsystems. We conclude the thesis with miscellaneous results on the divisor classnumber h, including a new approach for finding divisors of h.

Arithmetic Problems in Cubic and Quartic Function Fields