The ring of Fermat reals

We give the definition of the ring of Fermat reals, a simple extension of the real field containing nilpotent infinitesimals. The construction takes inspiration from smooth infinitesimal analysis, but provides a powerful theory of actual infinitesimals without any need of a background in mathematical logic. In particular it is consistent with classical logic. We face the problem to decide if the product of powers of nilpotent infinitesimals is zero or not, the identity principle for polynomials, the characterization of invertible elements and some applications to Taylor's formulas.