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

Computability in infinite Galois theory and algorithmically random algebraic fields