**Corollary 1**. If is a field, then every automorphism of is inner.

*Proof*. Let be an automorphism. Now let and apply the lemma we proved in part (1).

We now extend the lemma in part (1) to any finite dimensional central simple -algebra

**Lemma. **Let be a finite dimensional central simple -algebra and let be a simple -subalgebra of If is a -algebra homomorphism, then there exists an invertible element such that for all

*Proof*. We proved in here that for some integer Let and let By this corollary, is a simple -subalgebra of Now let

So is an -algebra homomorphism and thus, by the lemma in part (1), there exists an invertible element such that for all Let and choose Then

So Thus is in the centralizer of and hence by the second part of this lemma. Therefore

for some Note that since is invertible, is invertible too and Therefore for every we have

Thus

**Corollary 2**. (Skolem-Noether, 1929) Let be a finite dimensional central simple -algebra. Every automorphism of is inner. Also, if is a simple -subalgebra of and are -algebra homomorphisms, then there exists an invertible element such that for all

*Proof*. The first part is clear from the lemma. For the second part, we know from the lemma that there exist invertible elements such that and for all So if we let then for all