p-Adic ideals of p-rank d and the p-adic Nullstellensatz

In this paper we introduce and we study the notion of p-adic ideal of p-rank d and the one of p-adic radical of p-rank d of an ideal, by analogy with the notion of real ideal and the one of real radical of an ideal in the real case. Using those two notions and following more closely the real approach of Bochnak et al. (Géométrie algébrique réelle, Springer, Berlin 1987) we give a new proof of the p-adic Nullstellensatz.