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Quadratic fields of values of finite solvable groups
Abstract
We show that if G is a finite solvable group such that E = ℚ({χ(g) | χ∈Irr(G), g ∈ G}) is an extension of degree 2 over the rationals, then E = ℚ(√(± d)) for a positive integer d all of whose prime divisors lie in {2, 3, 5}. Moreover, if G has a composition factor of order 7, then either E = ℚ(√2) or E = ℚ(√(−3)). This provides a partial answer to a question of G. Navarro.
Related
- Ideal class groups of certain real quadratic number fields and factorizations of values of certain quadratic polynomials
- On Solvable Automorphism Groups of Function Fields over Finite Fields
- On a question of Navarro on the field of values of characters of solvable groups
- Finite abelian groups are Schur multipliers of finite solvable groups
- Automorphic symbols and automorphic \emph{L}-values of GL(2) over imaginary quadratic fields
BibTeX
@misc{debon2026quadratic,
title = {{Quadratic fields of values of finite solvable groups}},
author = {Sara C. Debón and Joan F. Tent},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.20992},
eprint = {2610.20992},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.20992},
}
RIS
TY - UNPB
TI - Quadratic fields of values of finite solvable groups
AU - Sara C. Debón
AU - Joan F. Tent
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.20992
UR - https://arxiv.org/abs/2610.20992
AB - We show that if G is a finite solvable group such that E = ℚ({χ(g) | χ∈Irr(G), g ∈ G}) is an extension of degree 2 over the rationals, then E = ℚ(√(± d)) for a positive integer d all of whose prime divisors lie in {2, 3, 5}. Moreover, if G has a composition factor of order 7, then either E = ℚ(√2) or E = ℚ(√(−3)). This provides a partial answer to a question of G. Navarro.
ER -