Integral generators in a certain quartic field and related diophantine equations
type: Item
instance of: scholarly article
title: Integral generators in a certain quartic field and related diophantine equations
zbMATH Open document ID: 0585.12005
author: Andrew Bremner
published in: Michigan Mathematical Journal
publication date: 1985-01-01T00:00:00Z
review text: 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.
reviewed by: Saburô Uchiyama
Mathematics Subject Classification ID: 11R23
Mathematics Subject Classification ID: 11R16
Mathematics Subject Classification ID: 11D25
zbMATH DE Number: 3937276
zbMATH Keywords: cubic diophantine equation
zbMATH Keywords: integral generators
zbMATH Keywords: quartic field
zbMATH Keywords: discriminants
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1029003240
OpenAlex ID: W2043372244
DOI: 10.1307/MMJ/1029003240
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description: scientific article; zbMATH DE number 3937276