From: Jyrki Lahtonen <lahtonen@utu.fi>
Subject: Re: Field and a proper subfield
Date: Tue, 02 Jan 2001 11:14:46 +0200
Newsgroups: sci.math
Summary: Luroth's theorem on unirational curves (subfields of  K(x) )

"David C. Ullrich" wrote:
> 
> On 31 Dec 2000 08:18:36 -0500, billpet@iol.com (Bill Petrank) wrote:
> 
> >Is there a field F that is isomorphic to a proper subfield of F ?
> 
> Yes. For example let F be the field of rational functions
> (quotients of polynomials) in countably many variables
> x_1, x_2, ... . (Rational functions with coefficients in
> whatever field you like.) Let F' be the subfield consisting
> of the functions that do not involve x_1. There's an
> obvious isomorphism from F to F'.
> 

You don't need countably many variables for this to work,
a single variable will suffice! For any field K, the field F=K(x)
(with x transcendental over K) has infinitely many subfields
isomorphic to the whole thing, say F'=K(x^2), F'=K(x^3),...

In fact Lüroth's theorem tells us that all the proper subfields
of F (properly containing K) are isomorphic to F

> >Thanks, Bill

Cheers,

Jyrki Lahtonen, Ph.D.
Department of Mathematics,
University of Turku,
FIN-20014 Turku, Finland

http://users.utu.fi/lahtonen