Harmonic analysis on a galois field and its subfields

Complex functions $\chi (m)$ where $m$ belongs to a Galois field $GF(p^ \ell)$, are considered. Fourier transforms, displacements in the $GF(p^ \ell) \times GF(p^ \ell)$ phase space and symplectic $Sp(2,GF(p^ \ell))$ transforms of these functions are studied. It is shown that the formalism inherits many features from the theory of Galois fields. For example, Frobenius transformations are defined which leave fixed all functions $h(n)$ where $n$ belongs to a subfield $GF(p^ d)$ of the $GF(p^ \ell)$. The relationship between harmonic analysis (or quantum mechanics) on $GF(p^ \ell)$ and harmonic analysis on its subfields, is studied.