> Also I think we have shown using a variant of Galois theory that there can not be a closed from solution for the nth prime.
Do you have a reference? This sounds interesting.
There are, of course, integer polynomials where the set of positive values is precisely the set of primes (think something like x²(y+1)-(z+x)y, but much more complicated) [2].
(This might seem like an interesting fact, but it really is not. All sets of numbers where a computer can decide whether a number belongs to the set have such a polynomial.)