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Dihedral fields with Bloch lattice index five and quadratic subfield Q(i)
Abstract
We construct infinitely many Galois extensions M/ℚ with dihedral group of order ten, quadratic subfield ℚ(i), and Bloch lattice index Q2(M)=5. More precisely, we specify a Chebotarev set of rational primes of density 1/50. For every prime ℓ in this set, at least two of four mixed cyclic quintic extensions of ℚ(i) of conductor (25ℓ) have this index and satisfy K2(OM)(5)≃(ℤ/5)2. A two-dimensional cup-product pencil selects the extensions. Generalized Bockstein maps, integral co-descent, and a motivic genus formula then determine the tame kernel and the subfield Bloch-lattice index. The result is unconditional.
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BibTeX
@misc{huimin2026dihedral,
title = {{Dihedral fields with Bloch lattice index five and quadratic subfield Q(i)}},
author = {Zheng Huimin},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.24043},
eprint = {2610.24043},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.24043},
}
RIS
TY - UNPB
TI - Dihedral fields with Bloch lattice index five and quadratic subfield Q(i)
AU - Zheng Huimin
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2610.24043
UR - https://arxiv.org/abs/2610.24043
AB - We construct infinitely many Galois extensions M/ℚ with dihedral group of order ten, quadratic subfield ℚ(i), and Bloch lattice index Q_2(M)=5. More precisely, we specify a Chebotarev set of rational primes of density 1/50. For every prime ℓ in this set, at least two of four mixed cyclic quintic extensions of ℚ(i) of conductor (25ℓ) have this index and satisfy K_2(O_M)(5)≃(ℤ/5)^2. A two-dimensional cup-product pencil selects the extensions. Generalized Bockstein maps, integral co-descent, and a motivic genus formula then determine the tame kernel and the subfield Bloch-lattice index. The result is unconditional.
ER -