  Let $F$ be a field of characteristic $0$ containing all roots of unity. We
construct a functorial compact Hausdorff space $X_F$ whose profinite
fundamental group agrees with the absolute Galois group of $F$, i.e. the
category of finite covering spaces of $X_F$ is equivalent to the category of
finite extensions of $F$.
  The construction is based on the ring of rational Witt vectors of $F$. In the
case of the cyclotomic extension of $\mathbb{Q}$, the classical fundamental
group of $X_F$ is a (proper) dense subgroup of the absolute Galois group of
$F$. We also discuss a variant of this construction when the field is not
required to contain all roots of unity, in which case there are natural
Frobenius-type automorphisms which encode the descent along the cyclotomic
extension.
