### Power integral bases in cubic relative extensions.

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Let $M$ be a given real quadratic field. We give a fast algorithm for determining all dihedral quartic fields $K$ with mixed signature having power integral bases and containing $M$ as a subfield. We also determine all generators of power integral bases in $K$. Our algorithm combines a recent result of Kable [9] with the algorithm of Gaál, Pethö and Pohst [6], [7]. To illustrate the method we performed computations for $M=\mathbb{Q}\left(\sqrt{2}\right),\mathbb{Q}\left(\sqrt{3}\right),\mathbb{Q}\left(\sqrt{5}\right).$

The problem of determining power integral bases in algebraic number fields is equivalent to solving the corresponding index form equations. As is known (cf. Győry [25]), every index form equation can be reduced to an equation system consisting of unit equations in two variables over the normal closure of the original field. However, the unit rank of the normal closure is usually too large for practical use. In a recent paper Győry [27] succeeded in reducing index form equations to systems of unit...

It is a classical problem in algebraic number theory to decide if a number field is monogeneous, that is if it admits power integral bases. It is especially interesting to consider this question in an infinite parametric family of number fields. In this paper we consider the infinite parametric family of simplest quartic fields $K$ generated by a root $\xi $ of the polynomial ${P}_{t}\left(x\right)={x}^{4}-t{x}^{3}-6{x}^{2}+tx+1$, assuming that $t>0$, $t\ne 3$ and ${t}^{2}+16$ has no odd square factors. In addition to generators of power integral bases we also calculate the minimal...

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