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

This proof, due to d’Alembert, relies on the following three facts:

Note that it suffices to prove that every polynomial with real coefficients has a complex root. Given a polynomial with complex coefficients, one can construct a polynomial with real coefficients by multiplying the polynomial by its complex conjugateMathworldPlanetmath. Any root of the resulting polynomial will either be a root of the original polynomial or the complex conjugate of a root.

The proof proceeds by inductionMathworldPlanetmath. Write the degree of the polynomial as 2n(2m+1). If n=0, then we know that it must have a real root. Next, assume that we already have shown that the fundamental theorem of algebraMathworldPlanetmath holds whenver n<N. We shall show that any polynomial of degree 2N(2m+1) has a complex root if a certain other polynomial of order 2N-1(2m+1) has a root. By our hypothesisMathworldPlanetmathPlanetmath, the other polynomial does have a root, hence so does the original polynomial. Hence, by induction on n, every polynomial with real coefficients has a complex root.

Let p be a polynomial of order d=2N(2m+1) with real coefficients. Let its factorization over the extension fieldMathworldPlanetmath E be


Next construct the d(d-1)/2=1 polynomials


where k is an integer between 1 and d(d-1)/2=1. Upon expanding the productMathworldPlanetmathPlanetmath and collecting terms, the coefficient of each power of x is a symmetric function of the roots ri. Hence it can be expressed in terms of the coefficients of p, so the coefficients of qk will all be real.

Note that the order of each qk is d(d-1)/2=2N-1(2m+1)(2N(2m+1)-1). Hence, by the induction hypothesis, each qk must have a complex root. By construction, each root of qk can be expressed as ri+rj+krirj for some choice of integers i and j. By the pigeonhole principle, there must exist integers i,j,k1,k2 such that both




are complex. But then ri and rj must be complex as well. because they are roots of the polynomial






Note.  D’Alembert was an avid supporter (in fact, the co-editor) of the famous French philosophical encyclopaedia. Therefore it is a fitting tribute to have his proof appear in the web pages of this encyclopaedia.


  • 1 Jean le Rond D’Alembert: “Recherches sur le calcul intégral”.   Histoire de l’Acadḿie Royale des Sciences et Belles Lettres, année MDCCXLVI, 182–224. Berlin (1746).
  • 2 R. Argand: “Réflexions sur la nouvelle théorie d’analyse”.  Annales de mathématiques 5, 197–209 (1814).
Title proof of fundamental theorem of algebra (due to d’Alembert)
Canonical name ProofOfFundamentalTheoremOfAlgebradueToDAlembert
Date of creation 2013-03-22 14:36:06
Last modified on 2013-03-22 14:36:06
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 10
Author rspuzio (6075)
Entry type Proof
Classification msc 30A99
Classification msc 12D99