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