On biquadratic fields that admit unit power integral basis
In number theory a main problem is that for which number fields there exists a power integral basis consisting of units. Let \(\mathbb K\) be a number field of degree \(n\) and \(\mathbb Z_{\mathbb K}\) be the maximal order of \(\mathbb K\). Furthermore assume \({\mathcal B} =\{1,\beta_2,\ldots , \beta_n \}\) is an integral basis of \(\mathbb Z_{\mathbb K}\) and let \(\eta = x_1 + x_2 \beta_2 +\dots + x_n \beta_n\in\mathbb Z_{\mathbb K}\). Then we consider the equation \[ I(\eta) = \left[\mathbb Z_{\mathbb K} : \mathbb Z[\Theta]\right] = F (x_2 , \ldots, x_n ) = \pm 1 \tag{1} \] which is solvable if and only if \(1, \Theta,\ldots , \Theta^{n-1}\) is a power integral bases of \(\mathbb K\). Now assume \(\Theta\in\mathbb Z_{\mathbb K}^*\) is a solution to the index equation (1). Then also \(\Theta^{-1}\) is a solution to (1). Beside ``trivial'' cases the solutions corresponding to \(\Theta\) and \(\Theta^{-1}\) should be distinct. Recently, this idea has been successfully applied in the case of pure quartic fields \(\mathbb K=\mathbb Q[\root 4 \of m]\). In this paper the authors consider biquadratic number fields whose maximal orders have power integral bases consisting of units. They prove an effective and efficient criteria to decide whether the maximal order of a biquadratic field has a unit power integral basis or not. In particular, they can determine all trivial biquadratic fields whose maximal orders have a unit power integral basis. At the end of this paper the authors apply one of their theorems to the family \[ \mathbb K=\mathbb Q\left(\sqrt{18n^2+17n+4},\sqrt{2n^2+n}\right) \] (that there is no unit power integral basis).
scientific article; zbMATH DE number 6006180
scientific article