proof of fundamental theorem of algebra (due to d’Alembert)
This proof, due to d’Alembert, relies on the following three facts:
For every polynomial with real coefficients, there exists a field in which the polynomial may be factored into linear terms. (For more information, see the entry “splitting field”.)
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 conjugate. 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 induction. Write the degree of the polynomial as . If , then we know that it must have a real root. Next, assume that we already have shown that the fundamental theorem of algebra holds whenver . We shall show that any polynomial of degree has a complex root if a certain other polynomial of order has a root. By our hypothesis, the other polynomial does have a root, hence so does the original polynomial. Hence, by induction on , every polynomial with real coefficients has a complex root.
Next construct the polynomials
where is an integer between and . Upon expanding the product and collecting terms, the coefficient of each power of is a symmetric function of the roots . Hence it can be expressed in terms of the coefficients of , so the coefficients of will all be real.
Note that the order of each is . Hence, by the induction hypothesis, each must have a complex root. By construction, each root of can be expressed as for some choice of integers and . By the pigeonhole principle, there must exist integers such that both
are complex. But then and 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)|
|Date of creation||2013-03-22 14:36:06|
|Last modified on||2013-03-22 14:36:06|
|Last modified by||rspuzio (6075)|