I know that $\displaystyle x^4 + y^4 = z^4$ has no obvious solutions in $\displaystyle \mathbb{Z}[i]$.

I know that $\displaystyle x^3 + y^3 = z^3$ has no obvious solutions in $\displaystyle \mathbb{Z}[\omega]$

I am curious, define $\displaystyle \zeta = e^{2\pi i/n}$.

Does $\displaystyle x^n + y^n = z^n $, $\displaystyle n\geq 2$, have ever solutions in $\displaystyle \mathbb{Z}[\zeta]$?

I ask this because I know that Kummer's dream of approaching FLT falls apart for some primes.