  We develop the analogue of the Witt construction in characteristic one. We
construct a functor from pairs of a perfect semi-ring of characteristic one and
an element strictly larger than one, to real Banach algebras. We find that the
entropy function familiar in thermodynamics, ergodic theory and information
theory occurs uniquely as the analogue of the Teichmuller polynomials in
characteristic one. We then apply the construction to the semi-field of
positive real numbers with max as addition, which plays a central role in
idempotent analysis and tropical geometry. Our construction gives the inverse
process of the ``dequantization" and provides a first hint towards an extension
of the field of real numbers relevant both in number theory and quantum
physics.
