  For an $n$-qubit system, a rational function on the space of mixed states
which is invariant with respect to the action of the group of local symmetries
may be viewed as a detailed measure of entanglement. We show that the field of
all such invariant rational functions is purely transcendental over the complex
numbers and has transcendence degree $4^n - 2n-1$. An explicit transcendence
basis is also exhibited.
