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PreprintCC BY 4.0arXiv

The module structure of ideals in degree p extensions of p-adic fields

  • Daniel Gil-Muñoz
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Abstract

Let L/K be a degree p extension of p-adic fields. We shall provide a complete characterization for the freeness of any fractional ideal pLh, h∈ℤ, of L as a module over its associated order AL/K(h) in the only Hopf-Galois structure on L/K, and an explicit description of AL/K(h). We show that there is no loss of generality in assuming that 0≤ h≤ p−1 and that the normal closure of L/K is totally ramified over K. If L/K is maximally ramified, then AL/K(h) is the maximal 𝒪K-order in H and pLh is AL/K(h)-free. The remaining cases correspond to typical extensions. For these, we provide an 𝒪K-basis of AL/K(h) and a characterization for the condition that pLh is AL/K(h)-free. In the non-exceptional cases, which had been partially studied in the Galois setting, the characterization depends on h and the continued fraction expansion of a/p, where a is the residue class mod p of the (fractional) ramification jump ℓ of L/K. In the remaining exceptional case, where L/K is almost maximally ramified and a+1≤ h≤ p−1, it consists in a numerical common minimizer condition.

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Cite

BibTeXRIS
BibTeX
@misc{gilmunoz2026module,
  title = {{The module structure of ideals in degree $p$ extensions of $p$-adic fields}},
  author = {Daniel Gil-Muñoz},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.02301},
  eprint = {2611.02301},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.02301},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - The module structure of ideals in degree p extensions of p-adic fields
AU  - Daniel Gil-Muñoz
PY  - 2026
DA  - 2026/11/01
PB  - arXiv
DO  - 10.48550/arxiv.2611.02301
UR  - https://arxiv.org/abs/2611.02301
AB  - Let L/K be a degree p extension of p-adic fields. We shall provide a complete characterization for the freeness of any fractional ideal p_L^h, h∈ℤ, of L as a module over its associated order A_L/K(h) in the only Hopf-Galois structure on L/K, and an explicit description of A_L/K(h). We show that there is no loss of generality in assuming that 0≤ h≤ p−1 and that the normal closure of L/K is totally ramified over K. If L/K is maximally ramified, then A_L/K(h) is the maximal 𝒪_K-order in H and p_L^h is A_L/K(h)-free. The remaining cases correspond to typical extensions. For these, we provide an 𝒪_K-basis of A_L/K(h) and a characterization for the condition that p_L^h is A_L/K(h)-free. In the non-exceptional cases, which had been partially studied in the Galois setting, the characterization depends on h and the continued fraction expansion of a/p, where a is the residue class mod p of the (fractional) ramification jump ℓ of L/K. In the remaining exceptional case, where L/K is almost maximally ramified and a+1≤ h≤ p−1, it consists in a numerical common minimizer condition.
ER  -