© 1996 by London Mathematical Society
© The London Mathematical Society
q-Linear Galois Theory
Université de Bordeaux I 351 cours de la libération, 33400 Talence, France E-mail thiery{at}math.u-bordeaux.fr
Received 21 June 1994. Revision received 21 November 1994.
Like elliptic curves, Drinfeld modules can be used to construct some representations of Galois groups. The initial purpose of this article is to give a well adapted ‘Galois theory’ to study these representations. The idea is to replace the minimal polynomial by a minimal 
q-linear polynomial because all polynomials involved in Drinfeld modules are q-linear. The multiplication must also be replaced by the action of the Frobenius map and the algebraic extensions by some finite dimensional vector spaces stable under the Frobenius map. To such new extension, 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 left ideals of this ring. The analogy with Galois theory is very deep and many important results can be proved: classification of unramified extensions of a complete field, local class field theory .... This so called q-linear Galois theory should have many interesting applications because most definitions of the classical Galois theory can be translated in this new language, and one can hope that this new approach will solve some old problems.