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Improving the Fundamental Theorem of Algebra

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Theorems 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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Correspondence to Joseph Shipman.

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Shipman, J. Improving the Fundamental Theorem of Algebra. The Mathematical Intelligencer 29, 9–14 (2007).

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  • Galois Group
  • Fundamental Theorem
  • Mathematical Intelligencer
  • Irreducible Polynomial
  • Splitting Field