Sur Le Groupe Des Extensions Cycliques

In this paper we give an explicit and elementary construction of the group of all cyclic extensions of prime degree of a given ring. We give explicit calculations in some particular cases. In the second part, under certains hypotheses concerning the base ring, we give a cohomological interpretation of this group. Next, the crucial question of the existence of primitive element for the algebra extensions of the vase ring is treated and finally, we examine certains cases where primitive elements always exists. © 1980.631268278Auslander, Goldman, The Brauer group of a commutative ring (1960) Transactions of the American Mathematical Society, 97, pp. 367-409Azumaya, On maximally central algebras (1951) Nagoya Math. J., 2, pp. 119-150Bourbaki, (1964) Algèbre Commutative, , Hermann, Paris, Chap. 5Chase, Harrison, Rosenberg, Galois theory and cohomology of commutative rings (1965) Mem. Amer. Math. Soc., 52, pp. 15-33Childs, The group of unramified Kummer extensions of prime degree (1977) Proceedings of the London Mathematical Society, 35, pp. 407-422Harrison, Abelian extensions of commutative rings (1965) Mem. Amer. Math. Soc., 52, pp. 1-14Knus, Ojanguren, Théorie de la descente et algèbres d'Azumaya (1974) Lecture Notes in Mathematics No. 389, , Springer-Verlag, Berlin/New YorkMicali, Villamayor, Sur les algèbres de Clifford, II (1970) J. Reine Angew. Math., 242, pp. 61-90Micali, Revoy, Modules quadratiques (1977) Cahiers Mathématiques n∘ 10, , Université de Montpellier II, MontpellierNakajima, On a group of cyclic extensions over commutative rings (1972) Math. J. Okayama Univ., 16, pp. 163-172Raynaud, Anneaux locaux henséliens (1960) Lecture Notes in Mathematics No. 169, , Springer-Verlag, Berlin/New YorkWang, Separable Algebras and Free Cubic Extensions Over Commutative Rings (1975) Ph. D. thesis, , Cornell UniversityWeiss, (1963) Algebraic Number Theory, , McGraw-Hill, New Yor