Primitive algebraic points on curves

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(Q̄) 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 Sd/2 S 2 ≀ S d / 2 .