Theorems of Bertini and Chevalley

TL;DR

This paper provides a concise proof of Chevalley's theorem on algebraic groups and discusses Bertini's irreducibility theorem, contributing to algebraic geometry theory.

Contribution

It offers a new, streamlined proof of Chevalley's theorem and addresses Bertini's irreducibility theorem within the same framework.

Findings

Confirmed that every algebraic group is an extension of an Abelian variety by a linear algebraic group

Provided a simplified proof of Chevalley's theorem

Discussed Bertini's irreducibility theorem

Abstract

We give a short proof of Chevalley's theorem that every algebraic group is an extension of an Abelian variety by a linear algebraic group. Along the way we treat Bertini's irreducibility theorem.