Joachim König

activity

20092026

most citedSeptic equations are solvable by 2-fold origami

2 citations · 4 across the 21 of their papers we have counts for

collaborators

27 papers

math.NT2026

The Davenport-Lewis-Schinzel problem on the reducibility of $f (X) - g (Y)$

Angelot Behajaina, Joachim König, Danny Neftin

We solve the problem of Davenport--Lewis--Schinzel (DLS), originating in the 1950s, regarding the reducibility of $f (X) - g (Y) \in C [X, Y]$ . This yields an almost-complete solut…

math.NT2025

A note on monogenic even polynomials

Joachim König

We extend several predecessor works on even sextic monogenic polynomials. In particular, we prove a conjecture of Lenny Jones, thereby classifying even sextic monogenic polynomials…

math.NT2025

Unramified extensions of quadratic number fields with Galois group $2. A_{n}$

Joachim König

We realize infinitely many covering groups $2. A_{n}$ (where $A_{n}$ is the alternating group) as the Galois group of everywhere unramified Galois extensions over infinitely many quadra…

math.NT2025

Integrally Hilbertian rings and the polynomial Schinzel hypothesis

Angelot Behajaina, Pierre Dèbes, Joachim König

The classical Hilbert specialization property is a field-theoretic tool ensuring that polynomial irreducibility over a field is preserved under specialization of some of the variab…

math.NT2024

Critical Point Criteria and Dynamically Monogenic Polynomials

Joachim König, Hanson Smith, Zack Wolske

Let $K$ be a number field with ring of integers $O_{K}$ , and let $f (x) \in O_{K} [x]$ be a monic, irreducible polynomial. We establish necessary and sufficient conditi…

math.NT2024

Linear constellations in primes with arithmetic restrictions

Christopher Frei, Joachim König, Magdaléna Tinková

We prove analogues of the theorem of Green and Tao on linear constellations in primes, in which the primes under consideration are restricted by certain arithmetic conditions. Our…

math.NT2024

Polynomial compositions with large monodromy groups and applications to arithmetic dynamics

Joachim König, Danny Neftin, Shai Rosenberg

For a composition $f = f_{1} \circ \dots \circ f_{r}$ of polynomials $f_{i} \in Q [x]$ of degrees $d_{i} \geq 5$ with alternating or symmetric monodromy group, we show that the monodromy…

math.NT2023

On sets of rational functions which locally represent all of $Q$

Benjamin Klahn, Joachim König

We investigate finite sets of rational functions ${f_{1}, f_{2}, \dots, f_{r}}$ defined over some number field $K$ satisfying that any $t_{0} \in K$ is a $K_{p}$ -value of one of…

math.NT2022

On fake subfields of number fields

Joachim König

We investigate the failure of a local-global principle with regard to "containment of number fields"; i.e., we are interested in pairs of number fields $(K_{1}, K_{2})$ such that $K_{2}$ …

math.NT2022

Quadratic number fields with unramified $S L_{2} (5)$ -extensions

Joachim König

Continuing the line of thought of an earlier work, we provide the first infinite family of quadratic number fields with everywhere unramified Galois extensions of Galois group $SL_…

math.NT2022

Locally cyclic extensions with Galois group $G L_{2} (p)$

Sara Arias-de-Reyna, Joachim König

Using Galois representations attached to elliptic curves, we construct Galois extensions of $Q$ with group $G L_{2} (p)$ in which all decomposition groups are cyclic. This is…

math.NT2021

Unramified extensions of quadratic number fields with Galois group $S L_{2} (7)$

Joachim König

We provide an infinite family of quadratic number fields with everywhere unramified Galois extensions of Galois group $S L_{2} (7)$ . To my knowledge, this is the first instance of infi…

math.NT2021

On parametric and generic polynomials with one parameter

Pierre Dèbes, Joachim König, François Legrand +1

Given fields $k \subseteq L$ , our results concern one parameter $L$ -parametric polynomials over $k$ , and their relation to generic polynomials. The former are polynomials $P(T,Y) \…

math.NT2020

Covering groups of $M_{22}$ as regular Galois groups over $Q$

Joachim König

We close a gap in the literature by showing that the full covering groups of the Mathieu group $M_{22}$ and of its automorphism group $A u t (M_{22})$ occur as the Galois group of $\m…

math.NT2020

Reducible Fibers of Polynomial Maps

Joachim König, Danny Neftin

For a degree $n$ polynomial $f$ over the rationals, the elements in the fiber $f^{- 1} (a)$ are of degree $n$ over $Q$ for most rational values $a$ by Hilbert's irreducibilit…

math.NT2019

Unramified extensions over low degree number fields

Joachim König, Danny Neftin, Jack Sonn

For various nonsolvable groups $G$ , we prove the existence of extensions of the rationals $Q$ with Galois group $G$ and inertia groups of order dividing $g e (G)$ , where $ge…

math.NT2018

On number fields with $k$ -free discriminants

Joachim König

Given a finite transitive permutation group $G$ , we investigate number fields $F / Q$ of Galois group $G$ whose discriminant is only divisible by small prime powers. This ge…

math.NT2017

The Grunwald problem and specialization of families of regular Galois extensions

Joachim König

We investigate specializations of infinite families of regular Galois extensions over number fields. The problem to what extent the local behaviour of specializations of one single…

math.GT2017★ 1 cited

Almost-Regular Dessins on a Sphere and Torus

Joachim König, Arielle Leitner, Danny Neftin

The Hurwitz problem asks which ramification data are realizable, that is appear as the ramification type of a covering. We use dessins d'enfant to show that families of genus 1 reg…

math.NT2017

On the local behaviour of specializations of function field extensions

Joachim König, François Legrand, Danny Neftin

Given a field $k$ of characteristic zero and an indeterminate $T$ over $k$ , we investigate the local behaviour at primes of $k$ of finite Galois extensions of $k$ arising as specia…

math.NT2016

Non-parametricity of rational translates of regular Galois extensions

Joachim König

We generalize a result of F.\ Legrand about the existence of non-parametric Galois extensions for a given group $G$ . More precisely, for a $K$ -regular Galois extension $F ∣ K (t)$ , we…

math.NT2016★ 1 cited

On the reducibility behaviour of Thue polynomials

Joachim König

We prove a result about reducibility behaviour of Thue polynomials over the rationals that was conjectured by Müller. More precisely, we show that, apart from few explicitly given…

math.AG2015★ 2 cited

Septic equations are solvable by 2-fold origami

Joachim König, Dmitri Nedrenco

In this paper we prove that a generic rational equation of degree $7$ is solvable by 2-fold origami. In particular we show how to septisect an arbitrary angle. This extends the wor…

math.NT2015

On rational functions with monodromy group $M_{11}$

Joachim König

We compute new polynomials with Galois group $M_{11}$ over $Q (t)$ . These polynomials stem from various families of covers of $P^{1} C$ ramified over at leas…

The Davenport-Lewis-Schinzel problem on the reducibility of f(X)−g(Y)

A note on monogenic even polynomials

Unramified extensions of quadratic number fields with Galois group 2.An​

Integrally Hilbertian rings and the polynomial Schinzel hypothesis

Critical Point Criteria and Dynamically Monogenic Polynomials

Linear constellations in primes with arithmetic restrictions

Polynomial compositions with large monodromy groups and applications to arithmetic dynamics

On sets of rational functions which locally represent all of Q

On fake subfields of number fields

Quadratic number fields with unramified SL2​(5)-extensions

Locally cyclic extensions with Galois group GL2​(p)

Unramified extensions of quadratic number fields with Galois group SL2​(7)

On parametric and generic polynomials with one parameter

Covering groups of M22​ as regular Galois groups over Q

Reducible Fibers of Polynomial Maps

Unramified extensions over low degree number fields

On number fields with k-free discriminants

The Grunwald problem and specialization of families of regular Galois extensions

Almost-Regular Dessins on a Sphere and Torus

On the local behaviour of specializations of function field extensions

Non-parametricity of rational translates of regular Galois extensions

On the reducibility behaviour of Thue polynomials

Septic equations are solvable by 2-fold origami

On rational functions with monodromy group M11​

The Davenport-Lewis-Schinzel problem on the reducibility of $f (X) - g (Y)$

Unramified extensions of quadratic number fields with Galois group $2. A_{n}$

On sets of rational functions which locally represent all of $Q$

Quadratic number fields with unramified $S L_{2} (5)$ -extensions

Locally cyclic extensions with Galois group $G L_{2} (p)$

Unramified extensions of quadratic number fields with Galois group $S L_{2} (7)$

Covering groups of $M_{22}$ as regular Galois groups over $Q$

On number fields with $k$ -free discriminants

On rational functions with monodromy group $M_{11}$