Algorithms for Galois Extensions of Global function Fields

Abstract

This thesis considers some computational problems in cyclic Galois extensions of global function fields. We investigate the efficient computation of integral closures, or maximal orders, in cyclic extensions of global fields and the determination of Galois groups for polynomials over global function fields.Global function fields, which are finite separable extensions of a global rational function field, are interesting because they provide a basis for designing efficient algorithms for algebraic curves. Applications of curves and function fields arise in coding theory and cryptography. Efficient computation with curves or function fields is necessary for efficient construction of codes.In this thesis, we consider function fields from the number theory point of view, and take advantage of algorithms for number fields (such as those in [2]), which can be used analogously for function fields. Algorithms are stated generally, so it is irrelevant whether the field is a number or a function field and whether or not it is represented as an extension of another algebraic (function) field.Some tasks have a rich history for number fields but have only recently sparked interest for function fields. These include the development of methods to efficiently compute integral closures (an analogue of Z), Galois groups, class groups and unit groups, which are the four most important tasks of number theory considered by Zassenhaus [6]. We investigate two of these. Integral closures can be used to compute class groups, unit groups and Galois groups, which are the other three important tasks.Also interesting for function fields is the computation of Riemann-Roch spaces of divisors and the genus. The calculation of a basis for a Riemann-Roch space can use the representation of a divisor by ideals of integral closures. Improving the efficiency

Galois groups over rational function fields and Explicit Hilbert Irreducibility

“…As noted in [7] Section 7.7, this algorithm can be adjusted to compute Galois groups of polynomials over fields other than the rational field. For example, [23,22] discusses this for polynomials over global rational and algebraic function fields. An algorithm for computing Galois groups of polynomials over Q(t) has been implemented in [6] and included in Magma V2.15.…”

“…We describe here some of the adjustments to the algorithm in [7] that are necessary for these computations. We address these adjustments using the same headings as [23,22] after providing a brief summary of the algorithm used. For a full exposition of the algorithm see [23] and [22], Algorithms 1 and 11, respectively.…”

“…We address these adjustments using the same headings as [23,22] after providing a brief summary of the algorithm used. For a full exposition of the algorithm see [23] and [22], Algorithms 1 and 11, respectively.…”

“…A starting group: (Step 2) Section 3.3 of [23,22] applies also for polynomials over Q(t). While all Galois groups are contained in S n , it can be more efficient if a smaller group containing the Galois group can be computed.…”

“…(1) Compute the subfields [9] of the field extension Q(t)[x] f and the Galois groups of the normal closures of these subfields. Invariants: (Step 3a) The invariants presented in [7] are sufficient here as all the rings involved have characteristic zero so there are no issues which arise in characteristic 2, as in [23,22].…”