  Let E/F be a CM field split above a finite place v of F, let l be a rational
prime number which is prime to v, and let S be the set of places of E dividing
lv. If E_S denotes a maximal algebraic extension of E unramified outside S, and
if u is a place of E dividing v, we show that any field embedding E_S \to
\bar{E_u} has a dense image.
  The "unramified outside S" number fields we use are cut out from the l-adic
cohomology of the "simple" Shimura varieties studied by Kottwitz and
Harris-Taylor. The main ingredients of the proof are then the local Langlands
correspondence for GL_n, the main global theorem of Harris-Taylor, and the
construction of automorphic representations with prescribed local behaviours.
  We explain how stronger results would follow from the knowledge of some
expected properties of Siegel modular forms, and we discuss the case of the
Galois group of a maximal algebraic extension of Q unramified outside a single
prime p and infinity.
