A glance into early attempts even to formulate the Fundamental Theorem of Algebra correctly, demonstrates how much development of mathematics has been impeded by the absence of a proper language to describe mathematical concepts.

The first formulation of the theorem was due to Albert Girard (1595-1632), a native of Lorraine who worked in Holland and was the editor of the works of Simon Stevin. His L'invention nouvelle en l'algèbre (Amsterdam, 1629) bases its fame on its formulation of the fundamental theorem of algebra, which also shows that he already took complex number seriously. It must be noted that Struik's is the only book that mentions Girard. The rest in my library that mention the Fundamental Theorem of Algebra, start with d'Alembert. The theorem has been for a long time known as the d'Alembert's Theorem. Here how Girard put it:

("Incomplete polynomials" are the ones in which some coefficients vanish. "Mixed" are, in my belief, the monomials a_kx^k.) He later wrote,

René Descartes (1596-1650), the father of Analytic Geometry, who made an attempt to formalize the whole of science by reducing it to the simplest and most obvious maxim "Cogito, ergo sum", still had troubles accepting negative numbers. In the following piece from his Géométrie the "false roots" he struggles with were unfortunate enough to need a minus sign in front of them.

The French mathematician Jean d'Alembert (1717-1783) tried but failed to prove the theorem in 1748. In the words of W.Dunham, d'Alembert recognized the importance of such a statement and made a stab at a proof. His stab, unfortunately, was wide of the mark. Perhaps to accord him the honor of trying, the result was long known as "d'Alembert's theorem", in spite of the fact that he came nowhere near proving it. This seems somewhat akin to renaming Moscow after Napoleon simply because he tried to reach it.

Euler made an attempt to prove the theorem in 1749 in a paper Recherches sur les racines imaginaires des équations. Gauss who opened his 1799 dissertation with a critical review of previous attempts, observed that Euler's proof lacks "the clarity which is required in mathematics." Euler proved a series of theorems for various polynomial degrees. First for n = 4, then n = 8, with special corollaries filling in the gaps. Then he stated the theorem for n = 2^k. The proof leads to an equation of degree N = 32870 which he calls "oddly even" (impairement pair) meaning that N/2 is odd.

Euler believed that the proof is solid ("je crois qu'on n'y trouvera rien à redire"), but to strengthen the argument he gives extra proofs for degrees 6, 4k + 2, 8k + 4,..., 2^kp, p an odd number. If Euler mixed a matter of belief with a matter of having proven something rigorously - who may not?

References

1. W.Dunham, Journey through Genius, Penguin Books, 1991
2. D.J.Struik, A Source Book in Mathematics, 1200-1800, Princeton University Press, Third Printing, 1990

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• Fundamental Theorem of Algebra: Statement and Significance
• What's in a proof?
• More about proofs
• Axiomatics
• Intuition and Rigor
• How to Prove Bolzano's Theorem
• Early attempts
• Proofs of the Fundamental Theorem of Algebra
  • Remarks on Proving The Fundamental Theorem of Algebra
    • Sketch of a Proof by Birkhoff and MacLane
    • Sketch of Second Proof (after Cauchy)
    • Details of the Cauchy's Proof
    • Note on the Extreme Value Theorem
    • Sketch of Proof by the methods of the theory of Complex Variables (after Liouville)
    • Maximum Modulus Theorem
  • A Proof of the Fundamental Theorem of Algebra: Standing on the shoulders of giants
  • Yet Another Proof of the Fundamental Theorem of Algebra
  • Fundamental Theorem of Algebra - Yet Another Proof
  • A topological proof, going in circles and counting