Efficient computation of maximal orders in radical (including Kummer) extensions

AbstractWe describe an algorithm, linear in the degree of the field, for computing a (pseudo) basis for P-maximal orders of radical (which includes Kummer) extensions of global arithmetic fields. We construct our basis in such a way as to further improve maximal order computations in these radical extensions. Using this algorithm for the similar problem of computing maximal orders of class fields is discussed. We give examples of both function fields and number fields comparing the running time of our algorithm to that of the Round 2 or 4 and Fraatz (2005)

Calculating the power residue symbol and ibeta: Applications of computing the group structure of the principal units of a p-adic number field completion

In the recent PhD thesis of Bouw, an algorithm is examined that computes the group structure of the principal units of a p-adic number field completion. In the same thesis, this algorithm is used to compute Hilbert norm residue symbols. In the present paper, we will demonstrate two other applications. The first application is the computation of an important invariant of number field completions, called ibeta. The algorithm that computes ibeta is deterministic and runs in polynomial time. The second application

Counting wildly ramified quartic extensions with prescribed discriminant and Galois closure group

Given a $2$-adic field $K$, we give formulae for the number of totally ramified quartic field extensions $L/K$ with a given discriminant valuation and Galois closure group. We use these formulae to prove a refinement of Serre's mass formula, which will have applications to the arithmetic statistics of number fields.Comment: Comments and corrections are welcome

On the computation of norm residue symbols

An algorithm is discussed to compute the exponential representation of principal units in a finite extension field F of the p-adic rationals. Also is discussed the computation of roots of unity contained in F and a special kind of principal unit, which is called a distinguished unit. The properties of norm residue symbols are given and also an algorithm to compute the norm residue symbol. Moreover a strongly distinguished unit is defined and an algorithm is given to compute such a unit. All the algorithms are polynomial time algorithms.Number theory, Algebra and Geometr

Algorithms for Galois extensions of global function fields

In this thesis we consider the computation of integral closures in cyclic Galois extensions of global function fields and the determination of Galois groups of polynomials over global function fields. The development of methods to efficiently compute integral closures and Galois groups are listed as two of the four most important tasks of number theory considered by Zassenhaus. We describe an algorithm each for computing integral closures specifically for Kummer, Artin--Schreier and Artin--Schreier--Witt extensions. These algorithms are more efficient than previous algorithms because they compute a global (pseudo) basis for such orders, in most cases without using a normal form computation. For Artin--Schreier--Witt extensions where the normal form computation may be necessary we attempt to minimise the number of pseudo generators which are input to the normal form. These integral closure algorithms for cyclic extensions can lead to constructing Goppa codes, which can correct a large proportion of errors, more efficiently. The general algorithm we describe to compute Galois groups is an extension of the algorithm of Fieker and Klueners to polynomials over function fields of characteristic p. This algorithm has no restrictions on the degrees of the polynomials it can compute Galois groups for. Previous algorithms have been restricted to polynomials of degree at most 23. Characteristic 2 presents additional challenges as we need to adjust our use of invariants because some invariants do not work in characteristic 2 as they do in other characteristics. We also describe how this algorithm can be used to compute Galois groups of reducible polynomials, including those over function fields of characteristic p. All of the algorithms described in this thesis have been implemented by the author in the Magma Computer Algebra System and perform effectively as is shown by a number of examples and a collection of timings

The unit residue group

The unit residue group, to which the present thesis is devoted, is defined using the norm-residue symbol, which Hilbert introduced into algebraic number theory in 1897. By its definition, the unit residue group of a global field is a direct sum of local contributions. It has a subgroup of a global nature, called the virtual group.We give a precise description of the unit residue groups and their virtual subgroups for some classes of number fields, including all quadratic fields. In addition we point out connections to two classical theorems on ideal class groups, namely the theorem of Armitage and Froehlich on 2-ranks and Scholz’s theorem on 3-ranks.We also study certain subgroups of the multiplicative group of a local field that play an important role in an algorithm for computing norm-residue symbols, and group isomorphisms between the groups of quadratic characters of two number fields that preserve L-series.Number theory, Algebra and Geometr