Integral generators in a certain quartic field and related diophantine equations (Q1071056)

Let \(\xi\) be a root of the quartic equation \(x^ 4-x+1=0\). \textit{T. Nagell} [Ark. Mat. 7, 359-394 (1968; Zbl 0164.352)] observes that the discriminants of \(\xi\), \(\xi^ 2\), \(\xi^ 3\), \(\xi^ 4\), \(\xi^ 6\), \(\xi^ 7\) are all equal to 229, and notes that it is not known if the discriminant of \(\xi^ m\) can equal 229, or equivalently \({\mathbb{Z}}[\xi^ m]={\mathbb{Z}}[\xi]\), for \(m>7\). In this paper the author proves that precisely the following powers of \(\xi\) generate the ring of integers \({\mathbb{Z}}[\xi]\) in \({\mathbb{Q}}(\xi):\) \(\xi^ m\) with \(\pm m\in \{1,2,3,4,6,7\}.\) It is shown more generally that, up to the equivalence mod \({\mathbb{Z}}\), precisely the following integers \(\alpha\) of \({\mathbb{Z}}[\xi]\) satisfy \({\mathbb{Z}}[\alpha]={\mathbb{Z}}[\xi]:\) \(\pm \alpha =\xi,\quad \xi^ 2,\quad \xi^ 3,\quad \xi \pm \xi^ 3,\quad \xi +\xi^ 2+\xi^ 3,\quad \xi^ 2\pm \xi^ 3,\quad \xi^ 2+2\xi^ 3,\quad 2\xi^ 2+\xi^ 3.\) The proof is achieved by determining the rational integer solutions (X,Y) of the diophantine equation \(X^ 3-4XY^ 2-Y^ 3=1\) (where \(x^ 3-4x- 1=0\) is the resolvent cubic equation associated to the quartic polynomial \(x^ 4-x+1)\), and involves considerable amount of arithmetical and numerical details about six particular quartic extensions of \({\mathbb{Q}}\) that occur in so doing. The proof is ingenious and the whole presentation is clear.