Introduction and Historical Remarks

  • Benjamin Fine
  • Gerhard Rosenberger
Part of the Undergraduate Texts in Mathematics book series (UTM)


The Fundamental Theorem of Algebra states that any complex polynomial must have a complex root. This basic result, whose first accepted proof was given by Gauss, lies really at the intersection of the theory of numbers and the theory of equations, and arises also in many other areas of mathematics. The purpose of these notes is to examine three pairs of proofs of the theorem. The first proof in each pair is fairly straightforward and depends only on what could be considered elementary mathematics. However, each of these first proofs lends itself to generalizations that in turn lead to more general results from which the Fundamental Theorem can be deduced as a direct consequence. These general results constitute the second proof in each pair.


Quadratic Formula Fundamental Theorem Field Extension Galois Theory Complex Root 
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Copyright information

© Springer Science+Business Media New York 1997

Authors and Affiliations

  • Benjamin Fine
    • 1
  • Gerhard Rosenberger
    • 2
  1. 1.Department of MathematicsFairfield UniversityFairfieldUSA
  2. 2.Department of MathematicsUniversity of DortmundDortmundGermany

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