Assume f(x)=x^(1/x). We are looking for it's inverse g(x)=f^-1(x) that can be expressed with a finite combination of the field operations (addition, multiplication, multiplicative inverse, addative inverse). We restrict ourselves to the rational numbers; since the rational numbers are closed under these operations it means that for each value of g(x), the argument of f(x) should be rational. However, taking g(2) we get 2=x^(1/x), for which no rational solution exists. We have a contradiction, which completes the proof.