On a Theorem by Hopf

We prove that the commutative nearly absolute valued division algebras have dimension ≤2. This extends a well-known theorem by H. Hopf asserting the same property for each commutative real division algebra of finite dimension. As an application we prove that R, C and C⋆ are the unique commutative real nearly absolute valued division algebras with norm satisfying ‖x2‖=‖x‖2.