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(2006 / vol.9 / no.1)
On power integral bases of the 2-elementary abelian extension
Yasuo Motoda, Toru Nakahara, and Kyoung Ho Park
Pages. 55-63     



Let $K$ be an abelian field whose Galois group is
2-elementary abelian over the rationals $Q.$ If $K$ is monogenic
and it is generated by a quadratic subfield
% $Q(sqrt{d_{1}m_{1}n_{1}ell})$ of $K$
and
a quartic subfield
% $Q(sqrt{mn},sqrt{dn})$
which are linearly disjoint, then $K$
coincides with the field $Q(sqrt{-1},sqrt{2},sqrt{-3}),$
namely $K$ is equal to the cyclotomic field $Q(zeta_{24})$ [MN].
In this article,
%we determine explicit integral bases of some real
%octic fields. Then
we prove that all the real and imaginary octic
fields $K$ are non-monogenic, namely the rings $Z_{K}$ of integers
in $K$ do not have any power integral basis except for
$Q(zeta_{24})$. Our method includes succinct new proofs for the
linearly disjoint case of [MN] and for Proposition 2 [PNM]
as a main tool of our purpose.



1. Introduction 2. The relative different 3. The case of $d_{1}m_{1}n_{1} geq 1$ 4. Problem



Integral basis, Relative different, Octic field, Monogenic field, Unit



11R04; Secondary: 11R11, 11R16