From: Torsten Ekedahl Subject: Re: Isomorphism of subfields Date: 22 Jan 2001 23:43:57 +0100 Newsgroups: sci.math.research Summary: Purely transcendental extensions of distinct fields can be isomorphic victor@idaccr.org (Victor S. Miller) writes: > I was recently asked the following question: > > If K and L are fields, x and y are transcendental over K and L > respectively, and K(x) is isomorphic to L(y), is K isomorphic to L? > > I suspect that in general the answer is no, but can't find a > counterexample. Does anyone either know a proof or a counterexample? When K is purely transcendtal over the complex numbers this is known as Zariski's problem that was negatively solved in Beauville, Arnaud; Colliot-Thélène, Jean-Louis; Sansuc, Jean-Jacques; Swinnerton-Dyer, Peter Variétés stablement rationnelles non rationnelles. (French) Ann. of Math. (2) 121 (1985), no. 2, 283--318. When the base field is not algebraically closed there were, if I remember correctly, counter examples prior to this.