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PreprintCC BY 4.0arXiv

Hilbert's Irreducibility for 𝔾m

  • Michael Stoll
  • Samir Siksek
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Abstract

Let K be a number field and S a finite set of non-archimedean places. Write π’ͺS for the ring of S-integers of K and π’ͺSΓ— for its unit group. Let Ο€: X β†’ β„™1 be a morphism of (irreducible) curves defined over K, and denote by Red(Ο€) the set of α∈ β„™1(K) such that the fibre Ο€βˆ’1(Ξ±) is reducible (i.e. the Galois action on the fibre is intransitive). Hilbert's Irreducibility Theorem asserts that Red(Ο€) is contained in a thin subset of β„™1(K). In this paper we give an explicit description of π’ͺSΓ— ∩ Red(Ο€). As an application we prove the following result inspired by a classical theorem of PΓ³lya and Siegel. Let p1,dotsc,ps be rational primes. Let f ∈ β„š[x].Then the following are equivalent: - There are infinitely many tuples (e1,dotsc,es) ∈ β„•s such that the polynomial f(x)βˆ’p1e1 β‹― pses is reducible. - f=p1a1 β‹― psas gβ„“ (with β„“ prime) or f=βˆ’4 p1a1 β‹― psas g4 for some g ∈ β„š[x] and some integers a1,dotsc,as.

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Cite

BibTeXRIS
BibTeX
@misc{stoll2026hilberts,
  title = {{Hilbert's Irreducibility for $\mathbbG_m$}},
  author = {Michael Stoll and Samir Siksek},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.04551},
  eprint = {2609.04551},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.04551},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Hilbert's Irreducibility for 𝔾_m
AU  - Michael Stoll
AU  - Samir Siksek
PY  - 2026
DA  - 2026/09/03
PB  - arXiv
DO  - 10.48550/arxiv.2609.04551
UR  - https://arxiv.org/abs/2609.04551
AB  - Let K be a number field and S a finite set of non-archimedean places. Write π’ͺ_S for the ring of S-integers of K and π’ͺ_S^Γ— for its unit group. Let Ο€: X β†’ β„™^1 be a morphism of (irreducible) curves defined over K, and denote by Red(Ο€) the set of α∈ β„™^1(K) such that the fibre Ο€^βˆ’1(Ξ±) is reducible (i.e. the Galois action on the fibre is intransitive). Hilbert's Irreducibility Theorem asserts that Red(Ο€) is contained in a thin subset of β„™^1(K). In this paper we give an explicit description of π’ͺ_S^Γ— ∩ Red(Ο€). As an application we prove the following result inspired by a classical theorem of PΓ³lya and Siegel. Let p_1,dotsc,p_s be rational primes. Let f ∈ β„š[x].Then the following are equivalent: - There are infinitely many tuples (e_1,dotsc,e_s) ∈ β„•^s such that the polynomial f(x)βˆ’p_1^e_1 β‹― p_s^e_s is reducible. - f=p_1^a_1 β‹― p_s^a_s g^β„“ (with β„“ prime) or f=βˆ’4 p_1^a_1 β‹― p_s^a_s g^4 for some g ∈ β„š[x] and some integers a_1,dotsc,a_s.
ER  -