Ring extensions of length two

Abstract

We characterize extensions of commutative rings [Formula: see text] such that [Formula: see text] is minimal for each [Formula: see text]-subalgebra [Formula: see text] of [Formula: see text] with [Formula: see text]. This property is equivalent to [Formula: see text] has length 2. Such extensions are either pointwise minimal or simple. We are able to compute the number of subextensions of [Formula: see text]. Besides commutative algebra considerations, our main result uses the concept of principal subfields of a finite separable field extension, which was recently introduced by van Hoeij et al. As a corollary of this paper, we get that simple extensions of length 2 have FIP.