Abstract
AbstractWe consider pure quartic relative extensions of the number field $${{\mathbb {Q}}}(i)$$
Q
(
i
)
of type $$K={{\mathbb {Q}}}(\root 4 \of {a+bi})$$
K
=
Q
(
a
+
b
i
4
)
, where $$a,b\in {{\mathbb {Z}}}$$
a
,
b
∈
Z
and $$b\ne 0$$
b
≠
0
, such that $$a+bi\in {{\mathbb {Z}}}[i]$$
a
+
b
i
∈
Z
[
i
]
is square-free. We describe integral bases of these fields. The index form equation is reduced to a relative cubic Thue equation over $${{\mathbb {Q}}}(i)$$
Q
(
i
)
and some corresponding quadratic form equations. We consider monogenity of K and relative monogenity of K over $${{\mathbb {Q}}}(i)$$
Q
(
i
)
. We shall show how our former method based on the factors of the index form can be used in the relative case to exclude relative monogenity in some cases.
Funder
Ministry of Innovation and Technology
Eötvös Lóránd Research Network
Publisher
Springer Science and Business Media LLC
Subject
Applied Mathematics,Analysis
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