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PreprintCC BY 4.0arXiv

Algorithms for p-rationality and for p-saturation of units

  • Tommy Hofmann
  • Henri Johnston
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Abstract

Let K be a number field. We give new practical algorithms that determine whether K is p-rational. For real cyclotomic fields K=ℚ(ζn)+, our method avoids computing either the class number or the full unit group. We also use the notion of p-rationality to explain the practical efficiency of an algorithm for determining whether a subgroup of the unit group 𝒪K× is p-saturated. This in turn yields a new algorithm for the unconditional verification of unit groups of number fields that substantially outperforms existing unconditional algorithms in practice. Finally, by combining our algorithms for determining p-rationality with work of Greenberg, we construct certain Galois representations ρ: Gal(ℚ̅/ℚ) → GLn(ℤp) with open image and further prescribed properties.

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Cite

BibTeXRIS
BibTeX
@misc{hofmann2026algorithms,
  title = {{Algorithms for $p$-rationality and for $p$-saturation of units}},
  author = {Tommy Hofmann and Henri Johnston},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.15190},
  eprint = {2609.15190},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.15190},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Algorithms for p-rationality and for p-saturation of units
AU  - Tommy Hofmann
AU  - Henri Johnston
PY  - 2026
DA  - 2026/09/14
PB  - arXiv
DO  - 10.48550/arxiv.2609.15190
UR  - https://arxiv.org/abs/2609.15190
AB  - Let K be a number field. We give new practical algorithms that determine whether K is p-rational. For real cyclotomic fields K=ℚ(ζ_n)^+, our method avoids computing either the class number or the full unit group. We also use the notion of p-rationality to explain the practical efficiency of an algorithm for determining whether a subgroup of the unit group 𝒪_K^× is p-saturated. This in turn yields a new algorithm for the unconditional verification of unit groups of number fields that substantially outperforms existing unconditional algorithms in practice. Finally, by combining our algorithms for determining p-rationality with work of Greenberg, we construct certain Galois representations ρ: Gal(ℚ̅/ℚ) → GL_n(ℤ_p) with open image and further prescribed properties.
ER  -