  We introduce a notion of algorithmic randomness for algebraic fields. We
prove the existence of a continuum of algebraic extensions of $\mathbb{Q}$ that
are random according to our definition. We show that there are noncomputable
algebraic fields which are not random. We also partially characterize the index
set, relative to an oracle, of the set of random algebraic fields computable
relative to that oracle.
  In order to carry out this investigation of randomness for fields, we develop
computability in the context of infinite Galois theory (where the relevant
Galois groups are uncountable), including definitions of computable and
computably enumerable Galois groups and computability of Haar measure on the
Galois groups.
