Mild pro-2-groups and 2-extensions of Q with restricted ramification

Using the mixed Lie algebras of Lazard, we extend the results of the first author on mild groups to the case p=2. In particular, we show that for any finite set S0 of odd rational primes we can find a finite set S of odd rational primes containing S0 such that the Galois group of the maximal 2-extension of Q unramified outside S is mild. We thus produce a projective system of such Galois groups which converge to the maximal pro-2-quotient of the absolute Galois group of Q unramified at 2 and ∞. Our results also allow results of Alexander Schmidt on pro-p-fundamental groups of marked arithmetic curves to be extended to the case p=2 over a global field which is either a function field of characteristic ≠2 or a totally imaginary number field.