Fundamental Theorem of Algebra: Proof

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SUMMARY

The discussion centers on an elementary proof of the Fundamental Theorem of Algebra (FTA) presented in a video. The proof, initially encountered in complex analysis, was deemed non-rigorous and primarily intuitive. The conversation highlights the connection between the FTA and Liouville's theorem, emphasizing that a rigorous proof of the FTA can be established using principles of continuity and convergence of sequences.

PREREQUISITES
  • Understanding of complex analysis principles
  • Familiarity with Liouville's theorem
  • Knowledge of continuity in mathematical functions
  • Concepts of convergence of sequences
NEXT STEPS
  • Study the rigorous proof of the Fundamental Theorem of Algebra
  • Explore Liouville's theorem in detail
  • Investigate continuity and its implications in complex analysis
  • Learn about convergence of sequences in mathematical analysis
USEFUL FOR

Mathematicians, students of complex analysis, and anyone interested in the rigorous foundations of algebraic theorems.

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TL;DR
Elementary proof of FTA
I found this video showing an elementary proof of the FTA.

 
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We were shown this approach in complex analysis, but it was said this argument is not rigorous and served only to give us an intuition, so we proceeded to prove the Liouville's theorem (bounded + entire implies constant iirc) and proved FTA as a consequence.

I haven't thought about this theorem for a while, I think the idea is correct and we should easily be able to make it rigorous, I mean I didn't see any leap of faith given everything we know about continuity and convergence of sequences etc. I might be missing something, though.
 
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