[math-fun] Question about a set

Consider the set S of numbers that can be gotten from elements of Q by addition, multiplication, and taking integer roots? That is, a in Q => a in S a, b in S => a + b in S a, b in S => ab in S a in R, k in Z, a^k in S => a in S I understand S would be a proper subset of the algebraic numbers. Does S have a name? Is S a field?