Iterative Derivations on Central Simple Algebras

Abstract

We prove that an iterative derivation $\hat{I}{´}_F$ on a field $F$ can be extended to an iterative derivation $\hat{I}{´}_A$ on a central simple $F-$algebra $A$ if the characteristic of $F$ does not divide the exponent of $A$ in the Brauer group of $F.$ For a central simple $F-$algebra with an iterative derivation, we show the existence of a unique (up to isomorphism) Picard-Vessiot splitting field and from the nature its Galois group, we also describe the structure of the central simple algebra in terms of its $\hat{I}{´}_A-$right ideals.