@unpublished{65358, abstract = {{We determine the asymptotic growth of extensions of local function fields of characteristic p counted by discriminant, where the Galois group is a subgroup of the affine group AGL_1(p). More general, we solve the corresponding counting problems for all groups which arise in a tower of a cyclic extension of order p over a cyclic extension of degree d coprime to p. This in particular give answers for certain non-abelian groups including S_3, dihedral groups of order 2p, and many Frobenius groups.}}, author = {{Klüners, Jürgen and Müller, Raphael}}, booktitle = {{arXiv:2604.02152}}, title = {{{Counting Frobenius extensions over local function fields}}}, year = {{2026}}, } @article{60874, abstract = {{Given a number field, it is an important question in algorithmic number theory to determine all its subfields. If the search is restricted to abelian subfields, one can try to determine them by using class field theory. For this, it is necessary to know the ramified primes. We show that the ramified primes of the subfield can be computed efficiently. Using this information we give algorithms to determine all the quadratic and the cyclic cubic subfields of the initial field. The approach generalises to cyclic subfields of prime degree. In the case of quadratic subfields, our approach is much faster than other methods.}}, author = {{Elsenhans, Andreas-Stephan and Klüners, Jürgen}}, issn = {{2772-8277}}, journal = {{Journal of Computational Algebra}}, publisher = {{Elsevier BV}}, title = {{{Computing quadratic subfields of number fields}}}, doi = {{10.1016/j.jaca.2025.100039}}, volume = {{15}}, year = {{2025}}, } @unpublished{55192, abstract = {{We describe the group of $\mathbb Z$-linear automorphisms of the ring of integers of a number field $K$ that preserve the set $V_{K,k}$ of $k$th power-free integers: every such map is the composition of a field automorphism and the multiplication by a unit. We show that those maps together with translations generate the extended symmetry group of the shift space $\mathbb D_{K,k}$ associated to $V_{K,k}$. Moreover, we show that no two such dynamical systems $\mathbb D_{K,k}$ and $\mathbb D_{L,l}$ are topologically conjugate and no one is a factor system of another. We generalize the concept of $k$th power-free integers to sieves and study the resulting admissible shift spaces.}}, author = {{Gundlach, Fabian and Klüners, Jürgen}}, booktitle = {{arXiv:2407.08438}}, title = {{{Symmetries of power-free integers in number fields and their shift spaces}}}, year = {{2024}}, } @article{55554, abstract = {{We discuss various connections between ideal classes, divisors, Picard and Chow groups of one-dimensional noetherian domains. As a result of these, we give a method to compute Chow groups of orders in global fields and show that there are infinitely many number fields which contain orders with trivial Chow groups.}}, author = {{Kirschmer, Markus and Klüners, Jürgen}}, issn = {{2522-0160}}, journal = {{Research in Number Theory}}, number = {{4}}, publisher = {{Springer Science and Business Media LLC}}, title = {{{Chow groups of one-dimensional noetherian domains}}}, doi = {{10.1007/s40993-024-00579-6}}, volume = {{10}}, year = {{2024}}, } @unpublished{57218, abstract = {{We arrange the orders in an algebraic number field in a tree. This tree can be used to enumerate all orders of bounded index in the maximal order as well as the orders over some given order.}}, author = {{Kirschmer, Markus and Klüners, Jürgen}}, booktitle = {{arXiv:2411.08568}}, title = {{{Enumerating orders in number fields}}}, year = {{2024}}, } @article{49372, author = {{Klüners, Jürgen and Wang, Jiuya}}, issn = {{2730-9657}}, journal = {{La Matematica}}, publisher = {{Springer Science and Business Media LLC}}, title = {{{Idélic Approach in Enumerating Heisenberg Extensions}}}, doi = {{10.1007/s44007-023-00067-w}}, year = {{2023}}, } @article{34839, abstract = {{We describe the relations among the ℓ-torsion conjecture, a conjecture of Malle giving an upper bound for the number of extensions, and the discriminant multiplicity conjecture. We prove that the latter two conjectures are equivalent in some sense. Altogether, the three conjectures are equivalent for the class of solvable groups. We then prove the ℓ-torsion conjecture for ℓ-groups and the other two conjectures for nilpotent groups.}}, author = {{Klüners, Jürgen and Wang, Jiuya}}, issn = {{0002-9939}}, journal = {{Proceedings of the American Mathematical Society}}, keywords = {{Applied Mathematics, General Mathematics}}, number = {{7}}, pages = {{2793--2805}}, publisher = {{American Mathematical Society (AMS)}}, title = {{{ℓ-torsion bounds for the class group of number fields with an ℓ-group as Galois group}}}, doi = {{10.1090/proc/15882}}, volume = {{150}}, year = {{2022}}, } @article{34835, abstract = {{We prove an upper bound for the asymptotics of counting functions of number fields with nilpotent Galois groups. }}, author = {{Klüners, Jürgen}}, issn = {{0065-1036}}, journal = {{Acta Arithmetica}}, keywords = {{Algebra and Number Theory}}, number = {{2}}, pages = {{165--184}}, publisher = {{Institute of Mathematics, Polish Academy of Sciences}}, title = {{{The asymptotics of nilpotent Galois groups}}}, doi = {{10.4064/aa211207-16-5}}, volume = {{204}}, year = {{2022}}, } @article{34840, abstract = {{In this paper we obtain a complete list of imaginary n-quadratic fields with class groups of exponent 3 and 5 under ERH for every positive integer n where an n-quadratic field is a number field of degree 2ⁿ represented as the composite of n quadratic fields. }}, author = {{Klüners, Jürgen and Komatsu, Toru}}, issn = {{0025-5718}}, journal = {{Mathematics of Computation}}, keywords = {{Applied Mathematics, Computational Mathematics, Algebra and Number Theory}}, number = {{329}}, pages = {{1483--1497}}, publisher = {{American Mathematical Society (AMS)}}, title = {{{Imaginary multiquadratic number fields with class group of exponent $3$ and $5$}}}, doi = {{10.1090/mcom/3609}}, volume = {{90}}, year = {{2021}}, } @article{34842, abstract = {{Let D<0 be a fundamental discriminant and denote by E(D) the exponent of the ideal class group Cl(D) of K=ℚ(√D). Under the assumption that no Siegel zeros exist we compute all such D with E(D) dividing 8. We compute all D with |D| ≤ 3.1⋅10²⁰ such that E(D) ≤ 8.}}, author = {{Elsenhans, Andreas-Stephan and Klüners, Jürgen and Nicolae, Florin}}, issn = {{0065-1036}}, journal = {{Acta Arithmetica}}, keywords = {{Algebra and Number Theory}}, number = {{3}}, pages = {{217--233}}, publisher = {{Institute of Mathematics, Polish Academy of Sciences}}, title = {{{Imaginary quadratic number fields with class groups of small exponent}}}, doi = {{10.4064/aa180220-20-3}}, volume = {{193}}, year = {{2020}}, } @article{34841, abstract = {{We give an exact formula for the number of G-extensions of local function fields Fq((t)) for finite abelian groups G up to a conductor bound. As an application we give a lower bound for the corresponding counting problem by discriminant. }}, author = {{Klüners, Jürgen and Müller, Raphael}}, issn = {{0022-314X}}, journal = {{Journal of Number Theory}}, keywords = {{Algebra and Number Theory}}, pages = {{311--322}}, publisher = {{Elsevier BV}}, title = {{{The conductor density of local function fields with abelian Galois group}}}, doi = {{10.1016/j.jnt.2019.11.007}}, volume = {{212}}, year = {{2020}}, } @article{34843, abstract = {{A polynomial time algorithm to find generators of the lattice of all subfields of a given number field was given in van Hoeij et al. (2013). This article reports on a massive speedup of this algorithm. This is primary achieved by our new concept of Galois-generating subfields. In general this is a very small set of subfields that determine all other subfields in a group-theoretic way. We compute them by targeted calls to the method from van Hoeij et al. (2013). For an early termination of these calls, we give a list of criteria that imply that further calls will not result in additional subfields. Finally, we explain how we use subfields to get a good starting group for the computation of Galois groups.}}, author = {{Elsenhans, Andreas-Stephan and Klüners, Jürgen}}, issn = {{0747-7171}}, journal = {{Journal of Symbolic Computation}}, keywords = {{Computational Mathematics, Algebra and Number Theory}}, pages = {{1--20}}, publisher = {{Elsevier BV}}, title = {{{Computing subfields of number fields and applications to Galois group computations}}}, doi = {{10.1016/j.jsc.2018.04.013}}, volume = {{93}}, year = {{2018}}, } @article{34844, abstract = {{Let k be a number field, K/k a finite Galois extension with Galois group G, χ a faithful character of G. We prove that the Artin L-function L(s,χ,K/k) determines the Galois closure of K over $\ℚ$. In the special case $k=\ℚ$ it also determines the character χ. }}, author = {{Klüners, Jürgen and Nicolae, Florin}}, issn = {{0022-314X}}, journal = {{Journal of Number Theory}}, keywords = {{Algebra and Number Theory}}, pages = {{161--168}}, publisher = {{Elsevier BV}}, title = {{{Are number fields determined by Artin L-functions?}}}, doi = {{10.1016/j.jnt.2016.03.023}}, volume = {{167}}, year = {{2016}}, } @article{34845, abstract = {{Computational Galois theory, in particular the problem of computing the Galois group of a given polynomial, is a very old problem. Currently, the best algorithmic solution is Stauduhar’s method. Computationally, one of the key challenges in the application of Stauduhar’s method is to find, for a given pair of groups H