6 On alpha - adic expansions in Pisot bases 1

We study α-adic expansions of numbers in an extension field, that is to say, left infinite representations of numbers in the positional numeration system with the base α, where α is an algebraic conjugate of a Pisot number β. Based on a result of Bertrand and Schmidt, we prove that a number belongs to Q(α) if and only if it has an eventually periodic α-expansion. Then we consider α-adic expansions of elements of the extension ring Z[α−1] when β satisfies the so-called Finiteness property (F). In the particular case that β is a quadratic Pisot unit, we inspect the unicity and/or multiplicity of α-adic expansions of elements of Z[α−1]. We also provide algorithms to generate α-adic expansions of rational numbers.