Harmonic Analysis on a Galois Field and Its Subfields

Complex functions χ(m) where m belongs to a Galois field GF(pℓ), are considered. Fourier transforms, displacements in the GF(pℓ)×GF(pℓ) phase space and symplectic 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 and Galois groups are introduced in the present context. The relationship between harmonic analysis on GF(pℓ) and harmonic analysis on its subfields, is studied.