On quartic unit group related with Ljunggren's Theorem
by
Ryotaro Okazaki
Doshisha University, Dept. Math., Kyotanabe, Kyoto, 610-0321 JAPAN

Ljunggren (1941) proved that x⁴-Dy²=1 has at most two positive integral solutions in x and y, where D is a non-square positive integer. A unit group of a quartic order appears in a difficult step of his proof. A clear description of the unit group shall be given. The step is as follows:

Let x > 1 be a given integer. Let \gamma, M in Z[i] satisfy

(x+i)\gamma2 - (x+1)M2 = -(1-i).

(Of course i denotes the imaginary unit.) We set

\xi = (1+i)(x+1)M2-1 + M\gamma\theta,    \lambda = (1+i)(x+1)-1+\theta,

where

\theta2 = 2(x+1)(x+i)i.

Let E the group of units of Z[i, \theta] whose relative norm to Z[i] is 1. The group E is generated by -1 and \lambda. In particular, we have

\xi = +/- \lambdan   for some odd n.

However, there is an example in which \xi is a square of a unit of Q(i, \theta).

On quartic unit group related with Ljunggren's Theorem

Date received: April 5, 1999

Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cacf-19.