A note on Wedderburn and Hasse theorems

Abstract

Wedderburn and Hasse theorems have always fascinated many algebraists as there are just very few versions of their proofs and that are somehow not easy to follow because they contain other results inside. In this note, we aim to look at their proofs again but nowwith more simplified versions that include sufficient details in order to let them be understandable for most readers in the subjects of finite fields and elliptic curves. In the proof of Wedderburn’s theorem, we succeed to show how it is equivalent to define either a stabilizer or centralizer of non-central elements and that helps to consider the equation on the order of group by using interchangeably the index of its subgroup and the order of the orbit. As for Hasse’s theorem, we provide a short proof in just two paragraphs with a recall of the most important results that have been used, and then, we state three of its version with an explanation of how this all leads in the end to the need of an algorithm like the one of René Schoof in 1985.