A Constructive Elementary Proof of the Skolem-Noether Theorem for Matrix Algebras

TL;DR

This paper provides a straightforward, elementary proof that all automorphisms of matrix algebras over a field are inner automorphisms, meaning they are conjugations by invertible matrices.

Contribution

It offers a constructive and elementary proof of the Skolem-Noether theorem specifically for matrix algebras, simplifying previous approaches.

Findings

Every K-automorphism of M_n(K) is conjugation by an invertible matrix.

The proof is elementary and constructive, avoiding complex algebraic machinery.

Confirms the inner automorphism nature of all automorphisms of matrix algebras.

Abstract

We give a constructive elementary proof for the fact that any K-automorphism of the full nxn matrix algebra over a field K is conjugation by some invertible nxn matrix A over K.