From: kramsay@aol.com (KRamsay) Subject: Re: Fermat in different rings. Date: 21 Mar 1999 17:37:24 GMT Newsgroups: sci.math In article , Nick Halloway writes: |I was wondering if anyone had made conjectures characterizing rings |that do have an FLT, say, which quadratic extensions of the rationals |do? Ribenboim's book has results about which quadratic extensions have nontrivial roots of X^n+Y^n=Z^n for specific values of n. For example, Q(sqrt(-7)) is the only one for n=4: (1+sqrt(-7))^4+(1-sqrt(-7))^4=2^4. "The proof is somewhat elaborate, but involves no real difficulty." For n=6 and n=9 there aren't any. For n=3 there are lots of them. I would guess that the quadratic extensions having no nontrivial solutions for n=3 mostly satisfy FLT, but that's just a guess, and I don't know offhand of any conjecture about it. Keith Ramsay