Newsgroups: sci.math
From: amlogan@flagstaff.princeton.edu (Adam M. Logan)
Subject: Re: PLSE HLP QUICK FIELD THRY? ??
Date: Mon, 20 Mar 1995 01:37:47 GMT

In article <Pine.SUN.3.91.950316135658.21597B-100000@math.uci.edu>,
Franisco Marquez  <fmarquez@math.uci.edu> wrote:
>
>I am tring to prove fields F1 and F2 are isomorphic so far i only
>have that there are monomorphism m1 and m2 such that 
>
>	m1:F1 -> F2 and
>	m2:F2 -> F1   
>in other words i can imbed each field in the other but i am not sure if 
>this is enough to conclude isomorphisms and if it is i don't see why?

(A solution to this problem has already been posted; here's an essentially
different one.)

Let f: E_1 \to E_2 be an isogeny of elliptic curves over, say, the complex
numbers, which is not a complex multiplication (in other words, E_1 and
E_2 are not isomorphic over C).  Let F_i be the function field of E_i.
Then f embeds the function field of E_2 into that of E_1.  The dual isogeny
of f (the isogeny such that, when composed on either side with f, gives
multiplication by n on the appropriate elliptic curve) embeds the function
field of E_1 into that of E_2.  But the function fields are not isomorphic,
because E_1 and E_2 are nonsingular projective curves over C which are
not isomorphic.

Adam