A number field K is primitive if K and \(\mathbb {Q}\) are the only subextensions of K. Let C be a curve defined over \(\mathbb {Q}\) . We call an algebraic point \(P\in C(\overline{\mathbb {Q}})\) primitive if the number field \(\mathbb {Q}(P)\) 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 \(C/\mathbb {Q}\) be a hyperelliptic curve of genus g, and let \(3 \le d \le g-1\) . 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 \(S_d\) or \(A_d\) . However, for any even \(d \ge 4\) , a hyperelliptic curve \(C/\mathbb {Q}\) has infinitely many imprimitive degree d points whose Galois group is a subgroup of \(S_2 \wr S_{d/2}\) .
