@article{34889, abstract = {{We prove that van Hoeij’s original algorithm to factor univariate polynomials over the rationals runs in polynomial time, as well as natural variants. In particular, our approach also yields polynomial time complexity results for bivariate polynomials over a finite field.}}, author = {{Belabas, Karim and van Hoeij, Mark and Klüners, Jürgen and Steel, Allan}}, issn = {{1246-7405}}, journal = {{Journal de Théorie des Nombres de Bordeaux}}, keywords = {{Algebra and Number Theory}}, number = {{1}}, pages = {{15--39}}, publisher = {{Cellule MathDoc/CEDRAM}}, title = {{{Factoring polynomials over global fields}}}, doi = {{10.5802/jtnb.655}}, volume = {{21}}, year = {{2009}}, } @inbook{35959, abstract = {{In this survey, we report about a new algorithm for factoring polynomials due to Mark van Hoeij. The main idea is that the combinatorial problem that occurs in the Zassenhaus algorithm is reduced to a very special knapsack problem. In case of rational polynomials, this knapsack problem can be very efficiently solved by the LLL algorithm. This gives a polynomial time algorithm, which also works very well in practice.}}, author = {{Klüners, Jürgen}}, booktitle = {{The LLL Algorithm}}, isbn = {{9783642022944}}, issn = {{1619-7100}}, publisher = {{Springer Berlin Heidelberg}}, title = {{{The van Hoeij Algorithm for Factoring Polynomials}}}, doi = {{10.1007/978-3-642-02295-1_8}}, year = {{2009}}, } @article{34895, abstract = {{We obtain strong information on the asymptotic behaviour of the counting function for nilpotent Galois extensions with bounded discriminant of arbitrary number fields. This extends previous investigations for the case of abelian groups. In particular, the result confirms a conjecture by the second author on this function for arbitrary groups in the nilpotent case. We further prove compatibility of the conjecture with taking wreath products with the cyclic group of order 2 and give examples in degree up to 8. }}, author = {{Klüners, Jürgen and Malle, G.}}, issn = {{0075-4102}}, journal = {{Journal für die reine und angewandte Mathematik (Crelles Journal)}}, keywords = {{Applied Mathematics, General Mathematics}}, number = {{572}}, pages = {{1--26}}, publisher = {{Walter de Gruyter GmbH}}, title = {{{Counting nilpotent Galois extensions}}}, doi = {{10.1515/crll.2004.050}}, volume = {{2004}}, year = {{2006}}, } @article{34891, abstract = {{We study the asymptotics conjecture of Malle for dihedral groups Dℓ of order 2ℓ, where ℓ is an odd prime. We prove the expected lower bound for those groups. For the upper bounds we show that there is a connection to class groups of quadratic number fields. The asymptotic behavior of those class groups is predicted by the Cohen--Lenstra heuristics. Under the assumption of this heuristic we are able to prove the expected upper bounds. }}, author = {{Klüners, Jürgen}}, issn = {{1246-7405}}, journal = {{Journal de Théorie des Nombres de Bordeaux}}, keywords = {{Algebra and Number Theory}}, number = {{3}}, pages = {{607--615}}, publisher = {{Cellule MathDoc/CEDRAM}}, title = {{{Asymptotics of number fields and the Cohen–Lenstra heuristics}}}, doi = {{10.5802/jtnb.561}}, volume = {{18}}, year = {{2006}}, } @article{34890, abstract = {{We prove that the 4-rank of class groups of quadratic number fields behaves as predicted in an extension due to Gerth of the Cohen–Lenstra heuristics. }}, author = {{Fouvry, Étienne and Klüners, Jürgen}}, issn = {{0020-9910}}, journal = {{Inventiones mathematicae}}, keywords = {{General Mathematics}}, number = {{3}}, pages = {{455--513}}, publisher = {{Springer Science and Business Media LLC}}, title = {{{On the 4-rank of class groups of quadratic number fields}}}, doi = {{10.1007/s00222-006-0021-2}}, volume = {{167}}, year = {{2006}}, } @inbook{35958, abstract = {{We establish a link between some heuristic asymptotic formulas (due to Cohen and Lenstra) concerning the moments of the p–part of the class groups of quadratic fields and formulas giving the frequency of the values of the p–rank of these class groups.}}, author = {{Fouvry, Étienne and Klüners, Jürgen}}, booktitle = {{Lecture Notes in Computer Science}}, isbn = {{9783540360759}}, issn = {{0302-9743}}, publisher = {{Springer Berlin Heidelberg}}, title = {{{Cohen–Lenstra Heuristics of Quadratic Number Fields}}}, doi = {{10.1007/11792086_4}}, year = {{2006}}, } @article{34892, abstract = {{We prove that the number of quartic S4--extensions of the rationals of given discriminant d is $O_\eps(d^{1/2+\eps})$ for all $\eps>0$. For a prime number p we derive that the dimension of the space of octahedral modular forms of weight 1 and conductor p or p² is bounded above by O(p¹/²log(p)²). }}, author = {{Klüners, Jürgen}}, issn = {{0065-1036}}, journal = {{Acta Arithmetica}}, keywords = {{Algebra and Number Theory}}, number = {{2}}, pages = {{185--194}}, publisher = {{Institute of Mathematics, Polish Academy of Sciences}}, title = {{{The number of S₄-fields with given discriminant}}}, doi = {{10.4064/aa122-2-3}}, volume = {{122}}, year = {{2006}}, } @article{34894, abstract = {{In this Note we give a counter example to a conjecture of Malle which predicts the asymptotic behavior of the counting functions for field extensions with given Galois group and bounded discriminant. }}, author = {{Klüners, Jürgen}}, issn = {{1631-073X}}, journal = {{Comptes Rendus Mathematique}}, keywords = {{General Mathematics}}, number = {{6}}, pages = {{411--414}}, publisher = {{Elsevier BV}}, title = {{{A counter example to Malle's conjecture on the asymptotics of discriminants}}}, doi = {{10.1016/j.crma.2005.02.010}}, volume = {{340}}, year = {{2005}}, } @article{34893, abstract = {{Let K be a global field and O be an order of K. We develop algorithms for the computation of the unit group of residue class rings for ideals O in . As an application we show how to compute the unit group and the Picard group of O provided that we are able to compute the unit group and class group of the maximal order O of K.}}, author = {{Klüners, Jürgen and Pauli, Sebastian}}, issn = {{0021-8693}}, journal = {{Journal of Algebra}}, keywords = {{Algebra and Number Theory}}, number = {{1}}, pages = {{47--64}}, publisher = {{Elsevier BV}}, title = {{{Computing residue class rings and Picard groups of orders}}}, doi = {{10.1016/j.jalgebra.2005.04.013}}, volume = {{292}}, year = {{2005}}, } @misc{42807, author = {{Klüners, Jürgen}}, isbn = {{978-3-8322-4003-5}}, pages = {{114}}, publisher = {{Shaker Verlag}}, title = {{{Über die Asymptotik von Zahlkörpern mit vorgegebener Galoisgruppe (Habilitation)}}}, year = {{2005}}, } @article{34896, abstract = {{We apply class field theory to the computation of the minimal discriminants for certain solvable groups. In particular, we apply our techniques to small Frobenius groups and all imprimitive degree 8 groups such that the corresponding fields have only a degree 2 and no degree 4 subfield.}}, author = {{Fieker, Claus and Klüners, Jürgen}}, issn = {{0022-314X}}, journal = {{Journal of Number Theory}}, keywords = {{Algebra and Number Theory}}, number = {{2}}, pages = {{318--337}}, publisher = {{Elsevier BV}}, title = {{{Minimal discriminants for fields with small Frobenius groups as Galois groups}}}, doi = {{10.1016/s0022-314x(02)00071-9}}, volume = {{99}}, year = {{2003}}, } @article{35954, abstract = {{Let {\ASIE K}\,/{\small \ℚ}({\ASIE t \!}) be a finite extension. We describe algorithms for computing subfields and automorphisms of {\ASIE K}\,/{\small \ℚ}({\ASIE t }\!). As an application we give an algorithm for finding decompositions of rational functions in {\small \ℚ(α)}. We also present an algorithm which decides if an extension {\ASIE L}\,/{\small \ℚ}({\ASIE t \!}) is a subfield of {\ASIE K}. In case [{\ASIE K : \;}{\small\ℚ}({\ASIE t \!})] = [{\ASIE L : \;}{\small \ℚ}({\ASIE t \!})] we obtain a {\small \ℚ}({\ASIE t \!})-isomorphism test. Furthermore, we describe an algorithm which computes subfields of the normal closure of {\ASIE K}\,/{\small \ℚ}({\ASIE t \!}).}}, author = {{Klüners, Jürgen}}, journal = {{Experiment. Math. }}, keywords = {{algorithms, decompositions, Galois groups, subfields}}, number = {{2}}, pages = {{171--181}}, publisher = {{Elsevier BV}}, title = {{{Algorithms for function fields}}}, volume = {{11}}, year = {{2002}}, } @article{34897, abstract = {{This paper announces the creation of a database for number fields. It describes the contents and the methods of access, indicates the origin of the polynomials, and formulates the aims of this collection of fields.}}, author = {{Klüners, Jürgen and Malle, Gunter}}, issn = {{1461-1570}}, journal = {{LMS Journal of Computation and Mathematics}}, keywords = {{Computational Theory and Mathematics, General Mathematics}}, pages = {{182--196}}, publisher = {{Wiley}}, title = {{{A Database for Field Extensions of the Rationals}}}, doi = {{10.1112/s1461157000000851}}, volume = {{4}}, year = {{2001}}, } @article{34900, abstract = {{We describe methods for the computation of Galois groups of univariate polynomials over the rationals which we have implemented up to degree 15. These methods are based on Stauduhar’s algorithm. All computations are done in unramified p -adic extensions. For imprimitive groups we give an improvement using subfields. In the primitive case we use known subgroups of the Galois group together with a combination of Stauduhar’s method and the absolute resolvent method.}}, author = {{Geissler, Katharina and Klüners, Jürgen}}, issn = {{0747-7171}}, journal = {{Journal of Symbolic Computation}}, keywords = {{Computational Mathematics, Algebra and Number Theory}}, number = {{6}}, pages = {{653--674}}, publisher = {{Elsevier BV}}, title = {{{Galois Group Computation for Rational Polynomials}}}, doi = {{10.1006/jsco.2000.0377}}, volume = {{30}}, year = {{2000}}, } @article{34901, abstract = {{Let L = K(α) be an Abelian extension of degree n of a number field K, given by the minimal polynomial of α over K. We describe an algorithm for computing the local Artin map associated with the extension L / K at a finite or infinite prime v of K. We apply this algorithm to decide if a nonzero a ∈ K is a norm from L, assuming that L / K is cyclic.}}, author = {{Acciaro, Vincenzo and Klüners, Jürgen}}, issn = {{0747-7171}}, journal = {{Journal of Symbolic Computation}}, keywords = {{Computational Mathematics, Algebra and Number Theory}}, number = {{3}}, pages = {{239--252}}, publisher = {{Elsevier BV}}, title = {{{Computing Local Artin Maps, and Solvability of Norm Equations}}}, doi = {{10.1006/jsco.2000.0361}}, volume = {{30}}, year = {{2000}}, } @article{34899, abstract = {{We describe methods for the construction of polynomials with certain types of Galois groups. As an application we deduce that all transitive groups G up to degree 15 occur as Galois groups of regular extensions of ℚ (t), and in each case compute a polynomial f ∈ ℚ [ x ] with Gal(f) = G.}}, author = {{Klüners, Jürgen and Malle, Gunter}}, issn = {{0747-7171}}, journal = {{Journal of Symbolic Computation}}, keywords = {{Computational Mathematics, Algebra and Number Theory}}, number = {{6}}, pages = {{675--716}}, publisher = {{Elsevier BV}}, title = {{{Explicit Galois Realization of Transitive Groups of Degree up to 15}}}, doi = {{10.1006/jsco.2000.0378}}, volume = {{30}}, year = {{2000}}, } @article{34898, abstract = {{We compute a polynomial with Galois group SL₂(11) over ℚ. Furthermore we prove that SL₂(11) is the Galois group of a regular extension of ℚ (t).}}, author = {{Klüners, Jürgen}}, issn = {{0747-7171}}, journal = {{Journal of Symbolic Computation}}, keywords = {{Computational Mathematics, Algebra and Number Theory}}, number = {{6}}, pages = {{733--737}}, publisher = {{Elsevier BV}}, title = {{{A Polynomial with Galois GroupSL2(11)}}}, doi = {{10.1006/jsco.2000.0380}}, volume = {{30}}, year = {{2000}}, } @article{34902, abstract = {{We present a new polynomial decomposition which generalizes the functional and homogeneous bivariate decomposition of irreducible monic polynomials in one variable over the rationals. With these decompositions it is possible to calculate the roots of an imprimitive polynomial by solving polynomial equations of lower degree.}}, author = {{Klüners, Jürgen}}, issn = {{0747-7171}}, journal = {{Journal of Symbolic Computation}}, keywords = {{Computational Mathematics, Algebra and Number Theory}}, number = {{3}}, pages = {{261--269}}, publisher = {{Elsevier BV}}, title = {{{On Polynomial Decompositions}}}, doi = {{10.1006/jsco.1998.0252}}, volume = {{27}}, year = {{1999}}, } @article{35941, abstract = {{Let L = ℚ(α) be an abelian number field of degree n. Most algorithms for computing the lattice of subfields of L require the computation of all the conjugates of α. This is usually achieved by factoring the minimal polynomial mα(x) of α over L. In practice, the existing algorithms for factoring polynomials over algebraic number fields can handle only problems of moderate size. In this paper we describe a fast probabilistic algorithm for computing the conjugates of α, which is based on p-adic techniques. Given mα(x) and a rational prime p which does not divide the discriminant disc(mα(x)) of mα(x), the algorithm computes the Frobenius automorphism of p in time polynomial in the size of p and in the size of mα(x). By repeatedly applying the algorithm to randomly chosen primes it is possible to compute all the conjugates of α.}}, author = {{Klüners, Jürgen and Acciaro, Vincenzo}}, issn = {{1088-6842}}, journal = {{Mathematics of Computation}}, number = {{227}}, pages = {{1179--1186}}, publisher = {{American Mathematical Society (AMS)}}, title = {{{Computing Automorphisms of Abelian Number Fields}}}, volume = {{68}}, year = {{1999}}, } @article{34905, abstract = {{Let ℚ(α) be an algebraic number field given by the minimal polynomial f of α. We want to determine all subfields ℚ(β) ⊂ Q(α) of given degree. It is convenient to describe each subfield by a pair (g, h) ∈ Z [t] x ℚ[t] such that g is the minimal polynomial of β = h(α) . There is a bijection between the block systems of the Galois group of f and the subfields of ℚ(α). These block systems are computed using cyclic subgroups of the Galois group which we get from the Dedekind criterion. When a block system is known we compute the corresponding subfield using p- adic methods. We give a detailed description for all parts of the algorithm.}}, author = {{Klüners, Jürgen}}, journal = {{Journal de Theorie des Nombres de Bordeaux}}, number = {{2}}, pages = {{243--271}}, publisher = {{Elsevier BV}}, title = {{{On computing subfields. A detailed description of the algorithm }}}, doi = {{https://jtnb.centre-mersenne.org/item/?id=JTNB_1998__10_2_243_0}}, volume = {{10}}, year = {{1998}}, }