Ring extensions of length 2

TL;DR

This paper characterizes length 2 extensions of commutative rings, showing they are either pointwise minimal or simple, and computes the number of subextensions, with implications for finite separable field extensions.

Contribution

It provides a complete characterization of length 2 ring extensions and links this to principal subfields in finite separable extensions, introducing new computational methods.

Findings

Length 2 extensions are either pointwise minimal or simple.

Number of subextensions of a length 2 extension can be computed.

Simple extensions of length 2 have FIP.

Abstract

We characterize extensions of commutative rings $R\subset S$ such that $R\subset T$ is minimal for each $R$-subalgebra $T$ of $S$ with $T\neq R,S$. This property is equivalent to $R\subset S$ has length 2. Such extensions are either pointwise minimal or simple. We are able to compute the number of subextensions of $R\subset S$. Besides commutative algebra considerations, our main result is a consequence of the recently introduced by van Hoeij et al. concept of principal subfields of a finite separable field extension. As a corollary of this paper, we get that simple extensions of length 2 have FIP.