TITLE: Fermat over Number Fields
QUESTION [6 upvotes]: If $n>2$, does the impossibility of solving $x^n +y^n=z^n$ with $x, y, z$ rational integers imply the same with $x, y, z$ algebraic integers?  
Rather, If insolvability in algebraic integers does follow, then does it follow from simple considerations, or is it still an interesting question?

REPLY [2 votes]: A part of the results of Kolyvagin was already proven by Peter Dénes,"Über den ersten Fall des letzten Fermatschen Satzes", Monatshefte für Mathematik, Bd. 54/3, 1950.
Dénes proved : let p be an odd prime, let K denote the p-th cyclotomic field, let P denote the prime ideal $(1 - \zeta)$ of K (where $\zeta$ denotes a primitive p-th root of unity); then in every solution (if any) of the equation $x^p + y^p + z^p = 0$ where x, y and z are P-integral elements of K not divisible by P, the rational integer t congruent to x/y (resp. y/z, resp. z/x) modulo P is a root of Kummer's system of congruences $B_{2i}l^{p-2i}(t + \zeta) \equiv 0 \pmod{p}$ for i = 1 to (p-3)/2, where $l^{j}$ denotes the j-th Kummer logarithmic function (with respect to $\zeta$).
It is Satz 1 of his paper, p. 162. You can read this paper here :
http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN362162050_0054&DMDID=DMDLOG_0017&LOGID=LOG_0017&PHYSID=PHYS_0166
The paper of Kolyvagin is here :
http://www.mathnet.ru/links/025935a4111ecc8ca5d60949a1020353/into103.pdf
The beginning of the English translation is here :
http://link.springer.com/article/10.1023/A%3A1017955309615#page-1
Kolyvagin's theorem 1 is the same as Bénes's Satz 1.
As you can see from the bibliography, Kolyvagin doesn't seem to know the paper of Dénes.