Relative Thue Equations

Let M ⊂ K be algebraic number fields, and let K = M(α) with an algebraic integer α. Let 0 ≠ μ ∈ ℤ M $$0\neq \mu \in {\mathbb Z}_M$$ . Consider the relative Thue equation Relative Thue equation Thue equations relative- N K ∕ M ( X − α Y ) = μ in X , Y ∈ ℤ M . $$\displaystyle N_{K/M}(X-\alpha Y)=\mu \;\;\; \mathrm {in} \;\;\; X,Y\in {\mathbb Z}_M. $$ Equations of this type were first considered in effective form by Kotov and Sprindzuk (Dokl Akad Nauk BSSR 17:393–395, 477, 1973). Kotov, S.V. Sprindžuk, V.G. This equation is a direct analogue of ( 3.1 ) in the relative case, when the ground ring is ℤ M $${\mathbb Z}_M$$ instead of ℤ $${\mathbb Z}$$ . The equation given in this form has only finitely many solutions. Relative Thue equations are often considered in the form N K ∕ M ( X − α Y ) = η μ in X , Y ∈ ℤ M , $$\displaystyle N_{K/M}(X-\alpha Y)=\eta \mu \;\;\; \mathrm {in}\;\;\; X,Y\in {\mathbb Z}_M, $$ where η is an unknown unit in M. In this case the solutions are determined up to a unit factor of M. In our applications index form equations can often be reduced to relative Thue equations for example for sextic fields (Sect. 11.2 ), octic fields (Sect. 14.2 ), nonic fields (Sect. 13.2 ), and for relative cubic (Sect. 13.1 ) and relative quartic extensions (Sect. 14.1 ).