Papers › Primitive algebraic points on curves
Primitive algebraic points on curves
Maleeha Khawaja, Samir Siksek
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A number field K is primitive if K and ℚ are the only subextensions of K. Let C be a curve defined over ℚ. We call an algebraic point P∈C(ℚ) primitive if the number field 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 be a hyperelliptic curve of genus g, and let 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 S_d or A_d. However, for any even d ≥4, a hyperelliptic curve C/ℚ has infinitely many imprimitive degree d points whose Galois group is a subgroup of S₂ ≀S_(d/2).
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