Algorithms in algebraic number theory

In this paper we discuss the basic problems of algorithmic algebraic number theory. The emphasis is on aspects that are of interest from a purely mathematical point of view, and practical issues are largely disregarded. We describe what has been done and, more importantly, what remains to be done in the area. We hope to show that the study of algorithms not only increases our understanding of algebraic number fields but also stimulates our curiosity about them. The discussion is concentrated of three topics: the determination of Galois groups, the determination of the ring of integers of an algebraic number field, and the computation of the group of units and the class group of that ring of integers.Comment: 34 page

Testing isomorphism of lattices over CM-orders

A CM-order is a reduced order equipped with an involution that mimics complex conjugation. The Witt-Picard group of such an order is a certain group of ideal classes that is closely related to the "minus part" of the class group. We present a deterministic polynomial-time algorithm for the following problem, which may be viewed as a special case of the principal ideal testing problem: given a CM-order, decide whether two given elements of its Witt-Picard group are equal. In order to prevent coefficient blow-up, the algorithm operates with lattices rather than with ideals. An important ingredient is a technique introduced by Gentry and Szydlo in a cryptographic context. Our application of it to lattices over CM-orders hinges upon a novel existence theorem for auxiliary ideals, which we deduce from a result of Konyagin and Pomerance in elementary number theory.Comment: To appear in SIAM Journal on Computin

Galois module structure of oriented Arakelov class groups

We show that Chinburg's Omega(3) conjecture implies tight restrictions on the Galois module structure of oriented Arakelov class groups of number fields. We apply our findings to formulating a probabilistic model for Arakelov class groups in families, offering a correction of the Cohen--Lenstra--Martinet heuristics on ideal class groups.Comment: 14 pages; comments welcom

On class groups of random number fields

The main aim of the present paper is to disprove the Cohen--Lenstra--Martinet heuristics in two different ways and to offer possible corrections. We also recast the heuristics in terms of Arakelov class groups, giving an explanation for the probability weights appearing in the general form of the heuristics. We conclude by proposing a rigorously formulated Cohen--Lenstra--Martinet conjecture.Comment: Expanded introduction, and other minor improvements in the exposition, 31 pages. Final version, to appear in Proc. London Math. So

Finding irreducible polynomials over finite fields

Wetensch. publicatieFaculteit der Wiskunde en Natuurwetenschappe

On Hats and other Covers

We study a game puzzle that has enjoyed recent popularity among mathematicians, computer scientist, coding theorists and even the mass press. In the game, n players are fitted with randomly assigned colored hats. Individual players can see their teammates ’ hat colors, but not their own. Based on this information, and without any further communication, each player must attempt to guess his hat color, or pass. The team wins if there is at least one correct guess, and no incorrect ones. The goal is to devise guessing strategies that maximize the team winning probability. We show that for the case of two hat colors, and for any value of n, playing strategies are equivalent to binary covering codes of radius one. This link, in particular with Hamming codes, had been observed for values of n of the form 2 m − 1. We extend the analysis to games with hats of q colors, q ≥ 2, where 1-coverings are not sufficient to characterize the best strategies. Instead, we introduce the more appropriate notion of a strong covering, and show efficient constructions of these coverings, which achieve winning probabilities approaching unity. Finally, we briefly discuss results on variants of the problem, including arbitrary input distributions, randomized playing strategies, and symmetric strategies

A family of exceptional polynomials in characteristic three

We present a family of indecomposable polynomials of non prime-power degree over the finite field of three elements which are permutation polynomials over infinitely many finite extensions of the field. The associated geometric monodromy groups are the simple groups PSL_2(3^k), where k>=3 is odd. This realizes one of the few possibilities for such a family which remain following the deep work of Fried, Guralnick and Saxl