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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 $ℂ$.

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 $\mathrm{sp}(A)$ 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 $ℂ$. 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 $ℂ$, hence it is $ℂ$.