Fields defined by polynomials

Abstract

αφ6 = P(α,6), aQb = Q(a,b), for all a and b in F yield a field (F, 0, 0)? It turns out that the answer is different for infinite fields than for finite fields, as shown in §§2 and 3. Next let R be the field of real numbers. For what quadruples Pu Pz> Qu Qz of real polynomials in four variables is (R x R, 0, 0) a field, when we set (α, 6) 0 (c, d) - (Pτ(a9 b, c, d), P2(a, b, c, d)) , (α, b) 0 (c, d) = (Q^α, 6, c, d), Q2(α, 6, c, d)) , where (x, y) denotes an ordered pair of real numbers? This question is partially answered in §§ 4 and 5, and in § 6 it is shown that the polynomials may be of arbitrarily high degree. In §7 it is proved that if definitions (II) do give a field, it must be isomorphic to the field of complex numbers.