  Let $L$ and $M$ be two algebraically closed fields contained in some common
larger field. It is obvious that the intersection $C=L\cap M$ is also
algebraically closed. Although the compositum $LM$ is obviously perfect, there
is no reason why it should be algebraically closed except when one of the two
fields is contained in the other. We prove that if the two fields are strictly
larger that $C$, and linearly disjoint over $C$, then the compositum $LM$ is
not algebraically closed; in fact we shall prove that the Galois group of the
maximal abelian extension of $LM$ is the free pro-abelian group of rank $|LM|$,
and that the free pro-nilpotent group of rank $|C|$ can be realized as a Galois
group over $LM$.
  The above results may be considered as the main contribution of this article
but we obtain some additional results on field composita that might be of
independent interest.
