Revisiting Kneser's Theorem for Field Extensions

TL;DR

This paper proves Hou's conjecture by extending Kneser's addition theorem to all field extensions, including non-separable cases, through an alternative proof and a strengthened theorem.

Contribution

It provides the first proof of Kneser's theorem for all field extensions, resolving Hou's conjecture and generalizing previous results to non-separable cases.

Findings

Theorem now valid for all field extensions, including non-separable.

Strengthening of Hou et al.'s theorem achieved.

Alternative proof methodology established.

Abstract

A Theorem of Hou, Leung and Xiang generalised Kneser's addition Theorem to field extensions. This theorem was known to be valid only in separable extensions, and it was a conjecture of Hou that it should be valid for all extensions. We give an alternative proof of the theorem that also holds in the non-separable case, thus solving Hou's conjecture. This result is a consequence of a strengthening of Hou et al.'s theorem that is a transposition to extension fields of an addition theorem of Balandraud.