Let p be a prime and let (mk)k<ω be a strictly increasing sequence of positive integers such that m0 = 1 and mk divides mk+1. A field F is said to be of type (p, (mk)k<ω) if it is the union of an increasing sequence (Fk)k<ω of fields such that Fk has pmk elements. A set X is called “finite” if it has n elements for some nonnegative integer n, and “Dedekind-finite” if every injection f: X → X is a bijection. If the Axiom of Choice is rejected, then it is relatively consistent to assume the existence of medial (that is, infinite, Dedekind-finite) sets. In this paper it is shown that given any type (p, (mk)k<ω) as above, it is relatively consistent with the usual axioms of set theory (minus Choice) to assume the existence of a medial field of type (p, (mk)k<ω). Conversely, it is shown that any medial field must be of type (p, (mk)k<ω) for some (p, (mk)k<ω) as above. The paper concludes with a few observations on Dedekind-finite rings. In the first part of the paper, a general knowledge of Fraenkel-Mostowski set theory and of the Jech-Sochor Embedding Theorems is assumed.