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Finally, two days ago, I succeeded - not on account of my hard efforts, but by the grace of the Lord. Like a sudden flash of lightning, the riddle was solved. I am unable to say what was the conducting thread that connected what I previously knew with what made my success possible.
In H. Eves Mathematical Circles Squared, Boston: Prindle, Weber and Schmidt, 1972.
Euler's Investigations on the Roots of Equations
Euler wrote Recherches sur les racines imaginaires des équations (Investigatons on the Imaginary Roots of Equations) while at the Berlin Academy, and it is found in the Mémoires de l'académie des sciences de Berlin, 1751, pages 222-288.
In the first part of this article, Euler concerns himself with what today we call the Fundamental Theorem of Algebra, or as Euler says in section 49,
Every rational function of a variable x, as
can always be resolved into real factors, either simple of the form x+p, or else double of the form xx+px+q.
Intimately related to this is the idea of complex numbers, which Euler treats in depth.
Euler works out the factorization for x4+2x3+4x2+2x+1 using clever, though accessible, algebra. Then he works out the factorization for a more general degree 4 equation. He discusses equations of odd and even degree, and shows how the number of real and imaginary factors relates to the parity of the degree of the equation. He continues by considering a large number of special cases, discussing each one in detail and relating them to each other.
Others have found that Euler did not completely sew up the matter in his proof, and indeed a complete proof of the Fundamental Theorem of Algebra that satisfies modern standards did not occur until over a century after this article was written.
Nevertheless, the reader will be well rewarded for following along as Euler works through this problem. There is much skillful algebra, and it is interesting to see basic results intermixed with more advanced manipulations. Euler is simply telling you what he is thinking.
Euler says the proof is complete in section 49, and we can perhaps detect a slight degree of unease when he writes that "in case one wanted to have trouble recognizing the correctness of these proofs, I am going to add several propositions concerning this subject that will not depend on the preceding, and whose truth will serve to lift any doubt that one might still have." Euler then offers additional proofs of some of the special cases.