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proof of fundamental theorem of algebra (due to d'Alembert)

Type of Math Object: 
Proof
Major Section: 
Reference
Groups audience: 

Mathematics Subject Classification

30A99 no label found12D99 no label found

Comments

I read about this wonderful proof in the book "Numbers" by Ebbinghaus et al. Since it's been a number of years since I lost my copy of this book, would someone out there who has access to a copy be kind enough to provide me with bibliographis information that I could add as a reference. Even better, maybe someone could provide a reference to the original publication by D'Alembert.

I think I have a copy at home. If no one else responds, I'll try to remember to send you something tonight (though a PM mail couldn't hurt to remind me)

Thanks,

Cam

Apparently, the proof was only really made rigorous by R. Argand in "Reflesions sur la nouvelle theorie d'analyse." (Annales de mathematiques 5, 197-209, 1814), but the idea of the proof can indeed be traced back to D'Alembert.

D'Alembert's proof, which Gauss said was not entirely satisfactory from a rigour point of view, can be found in "Recherches sue le calcul integral," in (Histoire de l'Academie Royale des Sciences et Belles Lettres), annee MDCCXLVI, Berlin 1748, 182-224)

Hope this helps,

Cam

Thank you for the references. I have added them to the entry.

The objection which Gauss and Argand raised was that D'Alembert's proof rests on the assumption that any polynomial has roots. Today, we understand that this is correct since, given a polynomial over a field, one can always construct an extension of the field in which the polynomial factors into linear factors. At the turn of the ninteteenth cenrury, the situation was a lot different. Mathematicians were still struggling to show that the complex number system was consistent (Argand planar representation of complex numbers went a good way towards reassuring mathematicians that complex numbers were legitimate) and abstract algebra was in a very rudimentary state. A good example of this is the fact that even the grandmaster Gauss had very little conception of group theory. In his disquisitions, Gauss demonstrates certain propositions about the composition of quadratic forms. Today, we would recognize them as the statement that quadratic forms form an Abelian group under composition, but, judging from the confused manner in which he explains what should be the statement of associativity, Gauss was not familiar with even the rudiments of group theory which any freshman of today would know.

It wasn't until the latter part of the ninetennth century that the necessary concepts of vector spaces, fields and the like were in place that Kronecker was able to show that splitting fields exist. This answered the objection to D'Alembert's proof (the proof of splitting fields in no way assumes the fundamental theorem of algebra, so there is no queestion of circularity) and we today can accept his proof as correct.

I think I will add a polished version of thease remarks as a historical postcript to my entry on D'Alembert's proof.

Thanks again,
Ray

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