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The exact Euclidean window of ℤ[√(14)]
Abstract
Let R = ℤ[√(14)], let π= 4 + √(14), and let p = (2,√(14)) = πR be the prime ideal above $2$. Cavallar and Lemmermeyer studied the multiplicative weighted norms ϕt(x) = |N(x)|(t/2)vπ(x) and asked for which weights t they are Euclidean functions on R. We show that the set of these weights, the Euclidean window of p, is (√5,T], where T is the algebraic number T = √(8/(104051√2 − 55617√7)) = 2.2468286690… = √5 + 0.0107606915…. Thus the window is nonempty but not open. We also obtain the integer-valued completely multiplicative Euclidean function 2vπ(x)|N(x)|6 on R.
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BibTeX
@misc{cardon2026exact,
title = {{The exact Euclidean window of $\mathbbZ[\sqrt14]$}},
author = {David A. Cardon},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.24309},
eprint = {2610.24309},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.24309},
}
RIS
TY - UNPB
TI - The exact Euclidean window of ℤ[√(14)]
AU - David A. Cardon
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2610.24309
UR - https://arxiv.org/abs/2610.24309
AB - Let R = ℤ[√(14)], let π= 4 + √(14), and let p = (2,√(14)) = πR be the prime ideal above $2$. Cavallar and Lemmermeyer studied the multiplicative weighted norms ϕ_t(x) = |N(x)|(t/2)^v_π(x) and asked for which weights t they are Euclidean functions on R. We show that the set of these weights, the Euclidean window of p, is (√5,T], where T is the algebraic number T = √(8/(104051√2 − 55617√7)) = 2.2468286690… = √5 + 0.0107606915…. Thus the window is nonempty but not open. We also obtain the integer-valued completely multiplicative Euclidean function 2^v_π(x)|N(x)|^6 on R.
ER -