1. Let F be any non-modular field, p an odd prime, P#1 a pth root of unity. Suppose that,u in F(?) is not the pth power of any quantity of F(r) so that the equation yp=,g is irreducible in F(r). Then the field F(y, h) is called a Kummert field over F. In the present paper we shall give a formal construction of all normal Kummer fields over F. This is equivalent to a construction of all fields F(x) of degree p over F such that F(x, t) is cyclic of degree p over F(?). In particular we provide a construction of all cyclic fields of degree p over F. We shall also apply the cyclic case to prove that a normal division algebra D of degree p over F is cyclic if and only if D contains a quantity y not in F such that yP=y in F. 2. The equation