On the Fundamental Theorem of Algebra

@article{Krner2006OnTF,
  title={On the Fundamental Theorem of Algebra},
  author={T. W. K{\"o}rner},
  journal={The American Mathematical Monthly},
  year={2006},
  volume={113},
  pages={347 - 348}
}
  • T. W. Körner
  • Published 1 April 2006
  • Mathematics, Philosophy
  • The American Mathematical Monthly
One of the simplest proofs that every nontrivial polynomial P has a zero goes as follows. Observe that |P(z)| → ∞ as |z| → ∞, so we may find an R > 0 with |P(z)| > |P(0)| for all |z| ≥ R. Since any real-valued continuous function on a compact set attains a minimum, |P(z)| attains a minimum for |z| ≤ R at some point z1 and, by the previous sentence, this must be a global minimum. By translation, we may suppose z1 = 0 and, by multiplying P by a constant, we may suppose that a0 = P(0) is real and… 
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