Survey on Bezout rings of p-adic analytic functions

International audienceLet K be a complete ultrametric algebraically closed field and let A(K) (resp. A(D)) be the K-algebra of analytic functions in K (resp. inside an open disk D). Following results in a paper by M. Lazard, we show that these algebras are Bezout rings, a property that is not showed in that paper. Moreover, the main results leading to the Bezout property is based upon a Mittag-Leffler theorem for meromorphic functions which is not proven in Lazard's paper. Furthermore, that Mittag-Leffler theorem (which is different from Krasner's Mittag-Leffler theorem for analytic elements) lets us find a shorter proof to show that the meromorphic functions admitting primitives are those whose residues are null.