On the monogenity of pure quartic relative extensions of \({{\mathbb {Q}}}(i)\)

We consider pure quartic relative extensions of the number field \({{\mathbb {Q}}}(i)\) of type \(K={{\mathbb {Q}}}(\root 4 \of {a+bi})\), where \(a,b\in {{\mathbb {Z}}}\) and \(b\ne 0\), such that \(a+bi\in {{\mathbb {Z}}}[i]\) is square-free. We describe integral bases of these fields. The index form equation is reduced to a relative cubic Thue equation over \({{\mathbb {Q}}}(i)\) and some corresponding quadratic form equations. We consider monogenity of K and relative monogenity of K over \({{\mathbb {Q}}}(i)\). 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.