Shimuras Reziprozitätsgesetz für den Körper der arithmetischen elliptischen Funktionen beliebiger Stufe.
Let be a finite extension of a global field. Such an extension can be generated over by a single element. The aim of this article is to prove the existence of a ”small” generator in the function field case. This answers the function field version of a question of Ruppert on small generators of number fields.
Let be a field whose characteristic is not and . We give a simple algorithm to find, given , a nontrivial solution in (if it exists) to the equation . The algorithm requires, in certain cases, the solution of a similar equation with coefficients in ; hence we obtain a recursive algorithm for solving diagonal conics over (using existing algorithms for such equations over ) and over .
En utilisant le théorème de Christol, Kamae, Mendès France et Rauzy, nous donnons une démonstration élémentaire de la transcendance de la série formelle ainsi que d’autres séries formelles à coefficients dans un corps fini.