PreprintCC BY 4.0arXiv
An infinite completely normal basis theorem
Abstract
Let L/K be an arbitrary Galois extension with group G, equipped with the Krull topology. Let C(G,K) denote the K-vector space of continuous functions G→ K, with K discrete and G acting by left translation. We prove that there exists a G-equivariant K-linear isomorphism φ:C(G,K)→ L such that, for every open subgroup H of G, the map φH:C(H,LH)→ L induced by extension by zero and extension of scalars is an H-equivariant LH-linear isomorphism. This gives a completely normal analogue of Lenstra's infinite normal basis theorem; the case of finite base fields was previously established in an equivalent form by Hachenberger.
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BibTeX
@misc{lundstrom2026infinite,
title = {{An infinite completely normal basis theorem}},
author = {Patrik Lundström},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.22247},
eprint = {2610.22247},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.22247},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - An infinite completely normal basis theorem
AU - Patrik Lundström
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.22247
UR - https://arxiv.org/abs/2610.22247
AB - Let L/K be an arbitrary Galois extension with group G, equipped with the Krull topology. Let C(G,K) denote the K-vector space of continuous functions G→ K, with K discrete and G acting by left translation. We prove that there exists a G-equivariant K-linear isomorphism φ:C(G,K)→ L such that, for every open subgroup H of G, the map φ^H:C(H,L^H)→ L induced by extension by zero and extension of scalars is an H-equivariant L^H-linear isomorphism. This gives a completely normal analogue of Lenstra's infinite normal basis theorem; the case of finite base fields was previously established in an equivalent form by Hachenberger.
ER -