124 research outputs found
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)
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
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Prime Decomposition in Iterated Towers and Discriminant Formulae
We explore certain arithmetic properties of iterated extensions. Namely, we compute the index associated to certain families of iterated polynomials and determine the decomposition of prime ideals in others
Finite Fields: Theory and Applications
Finite ďŹelds are the focal point of many interesting geometric, algorithmic and combinatorial problems. The workshop was devoted to progress on these questions, with an eye also on the important applications of ďŹnite ďŹeld techniques in cryptography, error correcting codes, and random number generation
On the essential logical structure of inter-universal TeichmĂźller theory in terms of logical AND ââ§â/logical OR ââ¨â relations: Report on the occasion of the publication of the four main papers on inter-universal TeichmĂźller theory
The main goal of the present paper is to give a detailed exposition of the essential logical structure of inter-universal TeichmĂźller theory from the point of view of the Boolean operators --such as the logical AND ââ§âlogical OR ââ¨â operators-- of propositional calculus. This essential logical structure of inter-universal TeichmĂźller theory may be summarized symbolically as follows: A ⧠B = A ⧠(Bâ â¨Ë Bâ â¨Ë...) â A ⧠(Bââ¨ËBââ¨Ë...â¨Ë BĚâ â¨Ë BĚâ â¨Ë...) -- where ¡ the ââ¨Ëâ denotes the Boolean operator exclusive-OR, i.e., âXORâ; ¡ A, B, Bâ, Bâ, BĚâ, BĚâ, denote various propositions; ¡ the logical AND ââ§'sâ correspond to the Î-link of inter-universal TeichmĂźller theory and are closely related to the multiplicative structures of the rings that appear in the domain and codomain of the Î-link; ¡ the logical XOR ââ¨Ë'sâ correspond to various indeterminacies that arise mainly from the log-Kummer-correspondence, i.e., from sequences of iterates of the log-link of inter-universal TeichmĂźller theory, which may be thought of as a device for constructing additive log-shells. This sort of concatenation of logical AND ââ§'sâ and logical XOR ââ¨Ë 'sâ is reminiscent of the well-known description of the âcarry-additionâ operation on TeichmĂźller representatives of the truncated Witt ring â¤/4⤠in terms of Boolean addition ââ¨Ëâ and Boolean multiplication ââ§â in the field Fâ and may be regarded as a sort of âBoolean intertwiningâ that mirrors, in a remarkable fashion, the âarithmetic intertwiningâ between addition and multiplication in number fields and local fields, which is, in some sense, the main object of study in inter-universal TeichmĂźller theory. One important topic in this exposition is the issue of âredundant copiesâ, i.e., the issue of how the arbitrary identification of copies of isomorphic mathematical objects that appear in the various constructions of inter-universal TeichmĂźller theory impacts-- and indeed invalidates-- the essential logical structure of inter-universal TeichmĂźller theory. This issue has been a focal point of fundamental misunderstandings and entirely unnecessary confusion concerning inter-universal TeichmĂźller theory in certain sectors of the mathematical community. The exposition of the topic of âredundant copiesâ makes use of many interesting elementary examples from the history of mathematics
Number Theory, Analysis and Geometry: In Memory of Serge Lang
Serge Lang was an iconic figure in mathematics, both for his own important work and for the indelible impact he left on the field of mathematics, on his students, and on his colleagues. Over the course of his career, Lang traversed a tremendous amount of mathematical ground. As he moved from subject to subject, he found analogies that led to important questions in such areas as number theory, arithmetic geometry and the theory of negatively curved spaces. Lang's conjectures will keep many mathematicians occupied far into the future.
In the spirit of Langâs vast contribution to mathematics, this memorial volume contains articles by prominent mathematicians in a variety of areas, namely number theory, analysis and geometry, representing Langâs own breadth of interests. A special introduction by John Tate includes a brief and engaging account of Serge Langâs life
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