While trying to solve a problem in number theory, I recently came across the concept of Fermat curves. This is the set of points in the complex projective plane, defined by $X^n+Y^n=Z^n$. In the Euclidean plane we may also speak of the set of solutions to $x^n+y^n=1$ as the Fermat curve. Often, if we're interested in the solutions over a field $K$, we might refer to this as the Fermat curve of degree $n$ over $K$.
The most famous result regarding these curves is Fermat's Last Theorem, which states that the Fermat curve of degree greater than $2$ has no nontrivial points over $\mathbb Q$. In light of this, I was wondering whether any work has been done studying Fermat curves over different subfields of $\mathbb R$, which seems like such an interesting problem given how famous FLT is, and the insights it has brought to the field of number theory. In particular, what work has been done over quadratic fields (of the form $\mathbb Q[\sqrt d]$)? Otherwise, is there a good reason why it hasn't been studied, due to a reason I'm unfamiliar of? I appreciate any insight!