On Quartic Thue Equations with Trivial Solutions

Abstract

Let K be a quartic number field with negative absolute discriminant and let ${\mathbf {L}} = {\mathbf {Q}}(\sqrt d )$ be its real quadratic subfield, with $d \equiv 3\;\pmod 4$. Moreover, assume K to be embedded in the reals. Further, let $\xi > 1$ generate the subgroup of units relative to L in the group of positive units of K. Under certain conditions, which can be explicitly checked, and for suitable linear forms $X(u,v)$ and $Y(u,v)$ with coefficients in ${\mathbf {Z}}[\sqrt d ]$, the diophantine equation \[ {\text {Norm}_{{\mathbf {K}}/{\mathbf {Q}}}}(X(u,v) + Y(u,v){\xi ^2}) = 1,\] which is a quartic Thue equation in the indeterminates u and v, has only trivial solutions, that is, solutions given by $XY = 0$. Information on a substantial number of equations of this type and their associated number fields is incorporated in a few tables.