Let K be a field of
characteristic p≠0. A field extension L∕K is said to split when there exist
intermediate fields J and D of L∕K where J is purely inseparable over K, D is
separable over K and L = J ⊗KD. K is modularly perfect if [K : Kp] ≦ p. Every
finitely generated extension of a modularly perfect field splits. This paper
develops criteria for an arbitrary extension L∕K to split and presents an
example of an extension of a modularly perfect field which does not split.
Necessary and/or sufficient conditions are also developed for the following
to hold for an extension L∕K: (a) L′∕K splits for every intermediate field
L′; (b) L′∕K is modular for every intermediate field L′; (c) L∕L′ splits for
every intermediate field L′; (d) L∕L′ is modular for every intermediate field
L′.
