Abstract
In this paper we characterize the monogenity of non-cyclic but abelian octic number fields \(L\) over the rationals \({{{\varvec{Q}}}},\) each of which is composed by a linearly disjoint cyclic quartic field of odd prime conductor \(\ell \) and a quadratic field of prime discriminant \(p^{*}.\) In the case of each odd conductor \(\ell |p^{*}|,\) the linear Diophantine equation with unit coefficients in a specified quadratic subfield of \(L\) is applied to determine the monogenity of the octic field \(L.\) In the case of each even conductor, among seven octic fields \(L\) of conductor \(5|2^{*}|,\) it is shown that the unknown four maximal imaginary subfields \(L\) of the \(40\)th cyclotomic field are non-monogenic, in which the non-monogenity of two fields are proved by the evaluation of the absolute norm of a partial factor \({\xi }-{\xi }^{{\rho }}\) of the different \(\mathfrak {d}({\xi })\) with a suitable Galois action \({\rho }\) for any integer \({\xi }\) in \(L\).
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Acknowledgments
The authors deeply thank to the referee and Prof. Lawrence Clinton Washington [Univ. of Maryland] for their valuable notices and comments. The authors also thank to Prof. Ken Yamamura for his Bibliography on monogenity of orders of algebraic number fields, July 2013, National Defense Academy of Japan updated ed. available on demand [118 papers with MR\(\#\) are included]. They are partially supported by Grant (\( \sharp 20540019\)) from the Japan Society for the Promotion of Science.
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Communicated by J. Schoißengeier.
To the memory of Professor Dr. Heinrich-Wolfgang Leopoldt.
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Sultan, M., Nakahara, T. On certain octic biquartic fields related to a problem of Hasse. Monatsh Math 176, 153–162 (2015). https://doi.org/10.1007/s00605-014-0670-y
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DOI: https://doi.org/10.1007/s00605-014-0670-y