A Note on the Field Isomorphism Problem of X3+sX+s and Related Cubic Thue Equations

Abstract

We study the field isomorphism problem of cubic generic polynomial X 3 þ sX þ s over the field of rational numbers with the specialization of the parameter s to nonzero rational integers m via primitive solutions to the family of cubic Thue equations x 3 À 2mx 2 y À 9mxy 2 À mð2m þ 27Þy 3 ¼ where 2 is a divisor of m 3 ð4m þ 27Þ 5 .