  Let K be an arithmetic function field, that is, a field of finite type over
the rational number field. In this note, as an application of the height theory
due to Chen-Moriwaki, we would like to show that the solutions of Fermat's
curve X^N + y^N = 1 of degree N over K consist of only either 0 or roots of
unity for almost positive integers N. More precisely, the density of such N is
1.
