Algorithmic aspects of biquadratic, cubic and radical function fields
Authors: Stephen V. Ullom
Venue:  (2007)

The thesis focuses on efficient computation of fundamental invariants of certain algebraic function fields, including biquadratic function fields, cubic function fields and radical function fields. Among the biquadratic function fields, Galois quartic function fields are discussed in more detail. Many problems in general function fields are mathematically or computationally hard to solve. Existing work focuses mainly on function fields of extension degree no more than 3 or on special types of fields such as superelliptic extensions [GPS02]. The thesis explores function fields of higher degree, such as biquadratic fields. These function fields involve relative quadratic extensions. We provide a comprehensive description of biquadratic function fields and their proper ties, including a characterization of the Galois and radical cases as well as the constant field. For the cyclic biquadratic scenario, by exploiting the interplay between the fundamental invariants of these relative quadratic extensions, we provide simple explicit formulas for the ratification index of any rational place, the field discriminant, the genus, and an algorithmically suitable integral basis. For the bicyclic biquadratic scenario, we compute the constant field, the signature and ultimately, the unit group of an arbitrary global biquadiatic bicyclic function field, using Kubota's method [Kub56]. We generalize Kubota's method to the function field scenario and a unified treatment is given for two different types of global fields. That is, computing the unit group of a bicyclic biquadratic global field K can be reduced to computing the three unit groups of the three distinct quadratic subfields of K. Infinite families of examples are also given for every possible type of unit group. These examples are independent of the constant field of K. Finally, we discuss two applications of the unit group computation; one involves the Minkowski unit in K and the other the norm of the fundamental unit in certain real quadratic function fields. For the cubic function field scenario, we present a method to compute the fundamental unit if the signature is (1, 1, 1, 2). This method can be generalized to any function field of arbitrary extension degree, if the signature of the function field is of the form (1,1,1, n - 1) and a neighbor calculation method exists. It must be pointed out that the latter requirement is the bottleneck of the unit group computation, and no satisfactory answer is obtained. We also study radical function fields which are more general than superelliptic ones. An explicit integral basis for a radical function field is constructed through a combinatorial method. The field discriminant is also obtained. We also study the P-signature for any place P of a radical function field and obtain certain formulae. We show that the residue degrees are not independent of the place lying above P through an example and that the P-signature is described uniquely when the radical function field is (cyclic) Galois over the rational function field. Above all, our work presents very explicit results, mostly stated as formulae and explicit representations, rather than algorithms. The formulae that we obtained for the ramifications and an integral basis of a cyclic biquadratic function field are simple, explicit and efficient. The computation of the unit group on a global bicyclic biquadratic function field generalizes and simplifies Kubota's work [Kub56) to function fields. It thus settles this question for all global bicyclic biquadratic extensions. The explicit construction of an integral basis of a radical function field is efficient and has a &ldquo;diagonal with denominators&rdquo; form, which is the simplest form that one can expect. This type of basis has no number field analogue.