  In this article we study the notion of essential subset of an additive basis,
that is to say the minimal finite subsets $P$ of a basis $A$ such that $A
\setminus P$ doesn't remains a basis. The existence of an essential subset for
a basis is equivalent for this basis to be included, for almost all elements,
in an arithmetic non-trivial progression. We show that for every basis $A$
there exists an arithmetic progression with a biggest common difference
containing $A$. Having this common difference $a$ we are able to give an upper
bound to the number of essential subsets of $A$: this is the radical's length
of $a$ (in particular there is always many finite essential subsets in a
basis).
  In the case of essential subsets of cardinality 1 (essential elements) we
introduce a way to "dessentialize" a basis. As an application, we definitively
improve the earlier result of Deschamps and Grekos giving an upper bound of the
number of the essential elements of a basis. More precisely, we show that for
all basis $A$ of order $h$, the number $s$ of essential elements of $A$ satisfy
$s\leq c\sqrt{\frac{h}{\log h}}$ where $c=30\sqrt{\frac{\log 1564}{1564}}\simeq
2,05728$, and we show that this inequality is best possible.
