Primitive algebraic points on curves

In this thesis we investigate various questions concerning rational points on curves defined over number fields.
Faltings’ finiteness theorem asserts that a curve defined over a number field K of genus greater than 1 has only finitely many points defined over K. This powerful result is ineffective and thus it is an interesting problem to determine all points on a curve defined over a fixed number field. In Chapter 3 we study the Fermat equation over real biquadratic fields, and moreover provide a complete resolution over the smallest (with respect to the discriminant) real biquadratic field. In Chapter 4 we study primitive algebraic points on curves defined over low degree number fields, and prove several sufficient sets of conditions for a curve to have finitely many primitive points of a fixed degree.