nt.number theory - Fermat over Number Fields

This is mostly an amplification of Kevin Buzzard's comment.

You ask about points on the Fermat curve $F_n: X^n + Y^n = Z^n$ with values in a number field $K$.

First note that since the equation is homogeneous, any nonzero solution with $(x,y,z) in K^3$ can be rescaled to give a nonzero solution $(Nx,Ny,Nz) in mathbb{Z}_K$, the ring of algebraic integers of $K$ -- here $N$ can taken to be an ordinary positive integer.

Thus you have two "parameters": the degree $n$ and the number field $K$.

If you fix $n$ and ask (as you have seemed to) whether there are solutions in some number field $K$, the answer is trivially yes as Kevin says: take $x$ and $y$ to be whatever algebraic integers you want; every algebraic integer has an $n$th root which is another algebraic integer, so you can certainly find a $z$ in some number field which gives a solution. Moreover, if you take $x$ and $y$ in a given number field $K$ (e.g. $mathbb{Q}$), then you can find infinitely many solutions in varying number fields $L/K$ of degrees at most $n$. But it is interesting to ask over which number fields (or which number fields of a given degree) there is a nontrivial solution.

On the other hand, if you fix the number field $K$ and ask for which $n$ the Fermat curve
$F_n$ has a solution $(x,y,z) in K$ with $xyz neq 0$, then you're back in business: this is a deep and difficult problem. (You can ask such questions for any algebraic curve, and many people, myself included, have devoted a large portion of their mathematical lives to this kind of problem.) So far as I know / remember at the moment, for a general $K$ there isn't that much which we know about this problem for the family of Fermat curves specifically, and there are other families (modular curves, Shimura curves) that we understand significantly better. But there are some beautiful general results of Faltings and Frey relating the plenitude of solutions (in fact not just over a fixed number field but over all number fields of bounded degree) to geometric properties of the curves, like the least degree of a finite map to the projective line (the "gonality").