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2 Corrected typo

I've lately have found myself admiring the proof of the fundamental theorem of algebra using linear algebra, due to H. Derksen, American Mathematical Monthly, 110 (7) (2003), 620–623.

He proves directly that linear operators on finite dimensional complex vector spaces admit eigenvectors, and deduces the fundamental theorem from this. I like the argument because it is completely elementary: All it uses is that odd dimensional polynomials over the reals have a real root, and that complex numbers have complex square roots (in particular, it avoids the machinery of complex or real analysis, and can even be presented without any reference to determinants). Moreover, the proof gives the result that $R(\sqrt{-1})$ is algebraically closed whenever $R$ is a real closed field, which before I had only seen proved using Galois theory or analogous, relatively sophisticated techniques.

Derksen's proof is a nice induction where first odd dimensions are taken care offof, then dimensions of the form 4k+2, then of the form 8k+4, etc.

He proves directly that linear operators on finite dimensional complex vector spaces admit eigenvectors, and deduces the fundamental theorem from this. I like the argument because it is completely elementary: All it uses is that odd dimensional polynomials over the reals have a real root, and that complex numbers have complex square roots (in particular, it avoids the machinery of complex or real analysis, and can even be presented without any reference to determinants). Moreover, the proof gives the result that $R(\sqrt{-1})$ is algebraically closed whenever $R$ is a real closed field, which before I had only seen proved using Galois theory or analogous, relatively sophisticated techniques.