Harmonic analysis on : Part I

The profinite group G ˆ which is the inverse limit of the inverse system comprised of the GF ( p p ℓ ) with homomorphisms between them, is studied. It is shown that Aut 0 ( G ˆ ) is a topological group isomorphic to Z p . The GF ( p p ∞ ) which is the direct limit of the direct system comprised of the GF ( p p ℓ ) with appropriate homomorphisms between them, is the Pontryagin dual of G ˆ . The Galois group of Frobenius automorphisms of GF ( p p ∞ ) (endowed with the Krull topology) is isomorphic to Z p . G ˆ can also be viewed as a topological right GF ( p p ∞ ) -module. The work provides the foundations for harmonic analysis on GF ( p p ∞ ) (and is continued in part II).