From: Torsten Ekedahl <teke@tofslan.matematik.su.se>
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.