Let
be a square-free integer. The aim of this paper is to construct
an integral basis of the pure octic field
L=ℚ
and to consider relative monogenity of
over
its quartic subfield
K=ℚ(∜m)
as well as over its quadratic subfield
K=ℚ(√m).
We prove that the field
is relatively monogenic over
for the
case of m ≡ 5,13 (mod 16) and does not have relative power
integral basis over
for m ≡ 1,9 (mod 16). Moreover
we prove that
has a relative power integral basis over
in the case of m ≡ 5, 9, 13 (mod 16). We show that the field ℚ
is
monogenic as well as relatively monogenic over
and
when m ≡ 2, 3 (mod 4).
In the case of
we prove our results by observing that the field
coincides with the 16th cyclotomic field
.