Computer algebra and, in particular, Gröbner bases are powerful tools in experimental design [G. Pistone and H. P. Wynn, Biometrika 83 (1996), no. 3, 653--666 MR1423881 ]. This paper applies this algebraic methodology to the identifiability of Fourier models. The choice of the class of trigonometric models forces one to deal with complex entities and algebraic irrational numbers. By means of standard techniques we have implemented a version of the Buchberger algorithm that computes Gröbner bases over the complex rational numbers and other simple algebraic extensions of the rational numbers. Some examples are fully worked out.