Improving the Fundamental Theorem of Algebra

@article{Shipman2007ImprovingTF,
  title={Improving the Fundamental Theorem of Algebra},
  author={Joseph C. Shipman},
  journal={The Mathematical Intelligencer},
  year={2007},
  volume={29},
  pages={9-14}
}
ConclusionTheorems 2 and 3 establish the minimum algebraic conditions necessary for a field to be algebraically closed, and they can therefore be said to “optimize” the Fundamental Theorem of Algebra. But each specific“degree implication” is a first-order consequence of the axioms for fields, and could have been discovered two centuries ago; the existence of these finitary relationships appears to have been unsuspected by practically everyone, with one important exception. 
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