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Improving the Fundamental Theorem of Algebra

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Conclusion

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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References

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    Conway, John H., “Effective Implications between the ‘Finite’ Choice Axioms,” inCambridge Summer School in Mathematical Logic (eds. A. R. D. Mathias, H. Rogers), Springer Lecture Notes in Mathematics 337, 439–458 (Springer-Verlag, Berlin 1971).

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    Dixon, John D., and Brian Mortimer,Permutation Groups, Springer Graduate Texts in Mathematics 163, Springer-Verlag, 1996.

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    Fine, Benjamin, and Gerhard Rosenberger,The Fundamental Theorem of Algebra, Springer-Verlag, New York 1997.

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    Gauss, Carl Friedrich,Werke, Volume 3, 33-56 (In Latin; English translation available at http://www.cs.man.ac.uk/~pt/misc/gauss-web.html).

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    Tarski, Alfred,A Decision Method for Elementary Algebra and Geometry, University of California Press, Berkeley and Los Angeles, 1951.

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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). https://doi.org/10.1007/BF02986170

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Keywords

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