F-Q-LINEAR GALOIS THEORY

Like elliptic curves, Drinfeld modules can be used to construct some r epresentations of Galois groups. The initial purpose of this article i s to give a well adapted 'Galois theory' to study these representation s. The idea is to replace the minimal polynomial by a minimal F-q-line ar polynomial because all polynomials involved in Drinfeld modules are F-q-linear. The multiplication must also be replaced by the action of the Frobenius map and the algebraic extensions by some finite dimensi onal vector spaces stable under the Frobenius map. To such new extensi on, one can associate the ring of its endomorphisms which commute with the Frobenius map. This is the analogue of the Galois group. The main theorem of this paper states a bijection between subextensions and le ft ideals of this ring. The analogy with Galois theory is very deep an d many important results can be proved: classification of unramified e xtensions of a complete field, local class field theory.... This so ca lled F-q-linear Galois theory should have many interesting application s because most definitions of the classical Galois theory can be trans lated in this new language, and one can hope that this new approach wi ll solve some old problems.