A field K algebraic over its
subfield F is said to be a J-extension (for Jónsson ω0-generated extension) of F if
K∕F is not finitely generated, but E∕F is finitely generated for each proper
intermediate field E. We seek to determine the structure of a given J-extension and
to determine the class of fields that admit a J-extension. Consideration of Galois
J-extensions plays a special role in each of these problems. In §2, we show that a
Galois extension K∕F is a J-extension if and only if Gal(K∕F) ≃Z∕pnZ for some
prime p. In §3, we show that F admits a J-extension if the algebraic closure
of F is infinite over F—that is, F is neither algebraically closed nor real
closed.
