Abstract
UDC 511
For any octic number field
K
generated by a root
α
of a monic irreducible trinomial
F
(
x
)
=
x
8
+
a
x
3
+
b
∈
ℤ
[
x
]
and for every rational prime
p
,
we show when
p
divides the index of
K
.
We also describe the prime power decomposition of the index
i
(
K
)
.
In this way, we give a partial answer to { Problem
22
} of Narkiewicz [<em>Elementary and analytic theory of algebraic numbers</em>, Springer-Verlag, Auflage (2004)] for this family of number fields. As an application of our results, we conclude that if
i
(
K
)
≠
1
,
then
K
is not monogenic. We illustrate our results by some computational examples.
Publisher
SIGMA (Symmetry, Integrability and Geometry: Methods and Application)
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