On quartic Thue equations with trivial solutions
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- by R. J. Stroeker PDF
- Math. Comp. 52 (1989), 175-187 Request permission
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.References
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Additional Information
- © Copyright 1989 American Mathematical Society
- Journal: Math. Comp. 52 (1989), 175-187
- MSC: Primary 11D25
- DOI: https://doi.org/10.1090/S0025-5718-1989-0946605-4
- MathSciNet review: 946605