A la mémoire de mon père.Abstract. Let p be a prime number, let K be a field of characteristic not p, containing the p-th roots of unity, and let r ≥ 1 be an integer. We compute the essential dimension of Z/ p r Z over K (Theorem 4.1). In particular, i) We have ed Q (Z/8Z) = 4, a result which was conjectured by Buhler and Reichstein in 1995 (unpublished).Acknowledgements. We thank Jean Fasel and Giordano Favi for fruitful discussions about essential dimension. We are also grateful to the referee for numerous comments which helped improve the clarity of the paper.

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The main result of this paper states that if k is a field of characteristic p > 0 and A/k is a central simple algebra of index d = p n and exponent p e , then A is split by a purely inseparable extension of k of the form k( p e √ a i , i = 1, . . . , d − 1). Combining this result with a theorem of Albert (for which we include a new proof), we get that any such algebra is Brauer equivalent to the tensor product of at most d − 1 cyclic algebras of degree p e . This gives a drastic improvement upon previously known upper bounds.

La simplicité est la réussite absolue. Après avoir joué une grande quantité de notes, toujours plus de notes, c'est la simplicité qui émerge, comme une récompense venant couronner l'art.

Abstract. Let X be a del Pezzo surface of degree 5 defined over a field F . A theorem of Yu. I. Manin and P. Swinnerton-Dyer asserts that every Del Pezzo surface of degree 5 is rational. In this paper we generalize this result as follows. Recall that del Pezzo surfaces of degree 5 over a field F are precisely the F -forms of the moduli space M 0,5 of stable curves of genus 0 with 5 marked points. Suppose n 5 is an integer, and F is an infinite field of characteristic = 2. It is easy to see that every twisted F -form of M 0,n is unirational over F . We show that (a) If n is odd, then every twisted F -form of M 0,n is rational over F . (b) If n is even, there exists a field extension F/k and a twisted F -form X of M 0,n such that X is not retract rational over F .