2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/71792Let F be an algebraically closed field with char(F) not equal to 2, let F/K be a Galois extension, and let X be a hyperelliptic curve defined over F. Let ιbe the hyperelliptic involution of X. We show that X can be defined over its field of moduli relative to the extension F/K if Aut(X)/<ι> is not cyclic. We construct explicit examples of hyperelliptic curves not definable over their field of moduli when Aut(X)/<ι> is cyclic.14 pagesNumber TheoryAlgebraic Geometry11G99;14H99text