 Hence we consider an arbitrary field formula_2 and its algebraic closure formula_4.An (imaginary) hyperelliptic curve of genus formula_5 over formula_2 is given by an equation of the form
formula_7 
where formula_8 is a polynomial of degree not larger than formula_5 and formula_10 is a monic polynomial of degree formula_11.Furthermore, we require the curve to have no singular points.