  Let 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
\iota be 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)/<\iota> is not
cyclic. We construct explicit examples of hyperelliptic curves not definable
over their field of moduli when Aut(X)/<\iota> is cyclic.
