Careful, none of these actually "set pi=1". Also, since pi is transcendental, the field would not be an algebraic number field, but in fact isomorphic to the r

Careful, none of these actually "set pi=1". Also, since pi is transcendental, the field would not be an algebraic number field, but in fact isomorphic to the rational functions over Q, i.e., quotients of polynomials with coefficients in Q.