nLab Gelfand-Mazur theorem

The Gel’fand–Mazur theorem states: the only complex Banach algebra which is also a field is the algebra of complex numbers $\mathbb{C}$.

The proof is a very simple consequence of the spectral theory of elements in a unital complex Banach algebra. It is a basic result of spectral theory that the spectrum $sp(A)$ of any element $a$ in a Banach algebra $A$ (which is by definition the set of complex numbers $\lambda$ such that $a- \lambda 1$ is not invertible) is a nonempty compact subset of $\mathbb{C}$. Now if the algebra is a field (or even a skewfield) then the only noninvertible element is $0$, hence every point in the spectrum of an arbitrary element $a\in A$ provides $\lambda$ such that $\lambda 1 = a$. Therefore the algebra can be identified with a unital complex subalgebra of $\mathbb{C}$, hence it is $\mathbb{C}$.