Consider the transcendental extension Q(t) of the field of rationals.
To Q(t) adjoin the root of the polynomial x^5+t^5=1. The resulting 
field Q(t)[x] is a radical extension of Q(t). Is it true that the 
only solutions to the equation x^n+y^n=1 in the field Q(t)[x] are 
(0,1), (1,0), (t,x), (x,t), (1/t,-x/t) and (-x/t, 1/t)?

Comment: Using the ABC theorem one can prove that the Fermat
curve x^n+y^n=1 does not have a non-trivial solution in Q(t) for n>2.
In particular in Q(t) the equation x^5+y^5=1 does not have
non-trivial solutions.