Abstract: By using Fermat’s method of descent, this paper proved that Diophantine equations { x^{4}-y^{4}=z^{2}} and { x^{4}+4y^{4}=z^{2}} have no non-trivial solutions over {\mathbf{Q}}(\sqrt{-3}), which implies that the Fermat Equation also has no non-trivial solutions in this field for n =4.

Key words: \quad ring of algebraic integers, \quad imaginary quadratic fields, Fermat’s method of descent, \quad ring of algebraic integers, \quad imaginary quadratic fields