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

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

•

Every polynomial with real coefficients which is of odd order has a real root. (This is a corollary of the intermediate value theorem.
•

Every second order polynomial with complex coefficients has two complex roots.
•

For every polynomial $p$ with real coefficients, there exists a field $E$ 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 $2^{n}(2m+1)$ . 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 algebra holds whenver $n<N$ . We shall show that any polynomial of degree $2^{N}(2m+1)$ has a complex root if a certain other polynomial of order $2^{N-1}(2m^{\prime}+1)$ has a root. By our hypothesis, 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 $d=2^{N}(2m+1)$ with real coefficients. Let its factorization over the extension field $E$ be

p(x)=(x-r_{1})(x-r_{2})\cdots(x-r_{d})

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

q_{k}(x)=\prod_{i<j}(x-r_{i}-r_{j}-kr_{i}r_{j})

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

Note that the order of each $q_{k}$ is $d(d-1)/2=2^{N-1}(2m+1)(2^{N}(2m+1)-1)$ . Hence, by the induction hypothesis, each $q_{k}$ must have a complex root. By construction, each root of $q_{k}$ can be expressed as $r_{i}+r_{j}+kr_{i}r_{j}$ for some choice of integers $i$ and $j$ . By the pigeonhole principle, there must exist integers $i,j,k_{1},k_{2}$ such that both

u=r_{i}+r_{j}+k_{1}r_{i}r_{j}

and

v=r_{i}+r_{j}+k_{2}r_{i}r_{j}

are complex. But then $r_{i}$ and $r_{j}$ must be complex as well. because they are roots of the polynomial

x^{2}+bx+c

where

b=-{k_{2}u+k_{1}v\over(k_{1}+k_{2})}

and

c={u-v\over k_{1}-k_{2}}

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.

References

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