# Two remarks on the inverse Galois problem for intersective polynomials

Jack Sonn[1]

• [1] Department of Mathematics Faculty of Mathematics Technion–Israel Institute of Technology Haifa, Israel
• Volume: 21, Issue: 2, page 435-437
• ISSN: 1246-7405

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## Abstract

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A (monic) polynomial $f\left(x\right)\in ℤ\left[x\right]$ is called intersective if the congruence $f\left(x\right)\equiv 0$ mod $m$ has a solution for all positive integers $m$. Call $f\left(x\right)$nontrivially intersective if it is intersective and has no rational root. It was proved by the author that every finite noncyclic solvable group $G$ can be realized as the Galois group over $ℚ$ of a nontrivially intersective polynomial (noncyclic is a necessary condition). Our first remark is the observation that the corresponding result for nonsolvable $G$ reduces to the ordinary inverse Galois problem for $G$ over $ℚ$. The second remark has to do with the scarcity of explicit examples of nontrivial intersective polynomials with given Galois group, and gives the first known example for the dihedral group of order ten.

## How to cite

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Sonn, Jack. "Two remarks on the inverse Galois problem for intersective polynomials." Journal de Théorie des Nombres de Bordeaux 21.2 (2009): 435-437. <http://eudml.org/doc/10890>.

@article{Sonn2009,
abstract = {A (monic) polynomial $f(x)\in \{\mathbb\{Z\}\}[x]$ is called intersective if the congruence $f(x)\equiv 0$ mod $m$ has a solution for all positive integers $m$. Call $f(x)$nontrivially intersective if it is intersective and has no rational root. It was proved by the author that every finite noncyclic solvable group $G$ can be realized as the Galois group over $\{\mathbb\{Q\}\}$ of a nontrivially intersective polynomial (noncyclic is a necessary condition). Our first remark is the observation that the corresponding result for nonsolvable $G$ reduces to the ordinary inverse Galois problem for $G$ over $\{\mathbb\{Q\}\}$. The second remark has to do with the scarcity of explicit examples of nontrivial intersective polynomials with given Galois group, and gives the first known example for the dihedral group of order ten.},
affiliation = {Department of Mathematics Faculty of Mathematics Technion–Israel Institute of Technology Haifa, Israel},
author = {Sonn, Jack},
journal = {Journal de Théorie des Nombres de Bordeaux},
language = {eng},
number = {2},
pages = {435-437},
publisher = {Université Bordeaux 1},
title = {Two remarks on the inverse Galois problem for intersective polynomials},
url = {http://eudml.org/doc/10890},
volume = {21},
year = {2009},
}

TY - JOUR
AU - Sonn, Jack
TI - Two remarks on the inverse Galois problem for intersective polynomials
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2009
PB - Université Bordeaux 1
VL - 21
IS - 2
SP - 435
EP - 437
AB - A (monic) polynomial $f(x)\in {\mathbb{Z}}[x]$ is called intersective if the congruence $f(x)\equiv 0$ mod $m$ has a solution for all positive integers $m$. Call $f(x)$nontrivially intersective if it is intersective and has no rational root. It was proved by the author that every finite noncyclic solvable group $G$ can be realized as the Galois group over ${\mathbb{Q}}$ of a nontrivially intersective polynomial (noncyclic is a necessary condition). Our first remark is the observation that the corresponding result for nonsolvable $G$ reduces to the ordinary inverse Galois problem for $G$ over ${\mathbb{Q}}$. The second remark has to do with the scarcity of explicit examples of nontrivial intersective polynomials with given Galois group, and gives the first known example for the dihedral group of order ten.
LA - eng
UR - http://eudml.org/doc/10890
ER -

## References

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1. D. Berend and Y. Bilu, Polynomials with roots modulo every integer. Proc. AMS 124 (1996), 1663–1671. Zbl1055.11523MR1307495
2. R. Brandl, Integer polynomials with roots mod $p$ for all primes $p$. J. Alg. 240 (2001), 822–835. Zbl1052.12002MR1841358
3. C. Jensen, A. Ledet, and N. Yui, Generic Polynomials. Cambridge Univ. Press, Cambridge-New-York, 2002. Zbl1042.12001MR1969648
4. J. Sonn, Polynomials with roots in ${ℚ}_{p}$ for all $p$. Proc. AMS 136 (2008), 1955–1960. Zbl1195.12007MR2383501

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