Complete solutions to a family of Thue equations of degree 12
Journal de théorie des nombres de Bordeaux, Volume 29 (2017) no. 2, pp. 549-568.

We consider a parametric non-Galois family of Thue equations ${F}_{m}\left(x,y\right)=\lambda$ of degree $12$ where $m$ is an integral parameter and $\lambda$ is a divisor of $729\left({m}^{2}+3m+9\right)$. Using the field isomorphism method which is developed in [15], we show that the equations have only the trivial solutions with $xy\left(x+y\right)\left(x-y\right)\left(x+2y\right)\left(2x+y\right)=0$.

We considérons une famille paramétrique non galoisienne d’équations de Thue ${F}_{m}\left(x,y\right)=\lambda$ de degré $12$$m$ est un paramètre entier et où $\lambda$ est un diviseur de $729\left({m}^{2}+3m+9\right)$. En utilisant la méthode d’isomorphismes de corps développée dans [15], nous montrons que ces équations ont seulement des solutions triviales avec $xy\left(x+y\right)\left(x-y\right)\left(x+2y\right)\left(2x+y\right)=0$.

Revised:
Accepted:
Published online:
DOI: 10.5802/jtnb.991
Classification: 11D25, 11D41, 11R16, 11R20, 12F10
Keywords: Thue equations, simplest cubic fields, simplest sextic fields.
Akinari Hoshi 1

1 Department of Mathematics Niigata University 8050 Ikarashi 2-no-cho, Nishi-ku, Niigata 950-2181, Japan
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Akinari Hoshi. Complete solutions to a family of Thue equations of degree 12. Journal de théorie des nombres de Bordeaux, Volume 29 (2017) no. 2, pp. 549-568. doi : 10.5802/jtnb.991. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.991/

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