  For positive integers $g$ and $N$, let $\mathcal{F}_N$ be the field of
meromorphic Siegel modular functions of genus $g$ and level $N$ whose Fourier
coefficients belong to the $N$th cyclotomic field. We present explicit
generators of $\mathcal{F}_N$ over $\mathcal{F}_1$ in terms of quotient of
theta constants, when $g\geq2$ and $N\geq 3$.
