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 , Franisco Marquez 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