Primitive algebraic points on curves

Abstract A number field K is primitive if K and Q Q are the only subextensions of K. Let C be a curve defined over Q Q. We call an algebraic point P C (Q) P ∈ C (Q ¯) primitive if the number field Q (P) Q (P) is primitive. We present several sets of sufficient conditions for a curve C to have finitely many primitive points of a given degree d. For example, let C/ Q C / Q be a hyperelliptic curve of genus g, and let 3 d g-1 3 ≤ d ≤ g - 1. Suppose that the Jacobian J of C is simple. We show that C has only finitely many primitive degree d points, and in particular it has only finitely many degree d points with Galois group Sd S d or Ad A d. However, for any even d 4 d ≥ 4, a hyperelliptic curve C/ Q C / Q has infinitely many imprimitive degree d points whose Galois group is a subgroup of S₂ S₃/₂ S 2 ≀ S d / 2.