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 $$. We shall show that any polynomial of degree ${2}^{N}(2m+1)$ has a complex root if a certain other polynomial of order ${2}^{N1}(2{m}^{\prime}+1)$ has a root. By our hypothesis^{}, the other polynomial does have a root, hence so does the original polynomial. Hence, by induction on $n$, every polynomial with real coefficients has a complex root.
Let $p$ be a polynomial of order $d={2}^{N}(2m+1)$ with real coefficients. Let its factorization over the extension field^{} $E$ be
$$p(x)=(x{r}_{1})(x{r}_{2})\mathrm{\cdots}(x{r}_{d})$$ 
Next construct the $d(d1)/2=1$ polynomials
$$ 
where $k$ is an integer between $1$ and $d(d1)/2=1$. 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 $d(d1)/2={2}^{N1}(2m+1)({2}^{N}(2m+1)1)$. Hence, by the induction hypothesis, each ${q}_{k}$ must have a complex root. By construction, each root of ${q}_{k}$ can be expressed as ${r}_{i}+{r}_{j}+k{r}_{i}{r}_{j}$ 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
$$b=\frac{{k}_{2}u+{k}_{1}v}{({k}_{1}+{k}_{2})}$$ 
and
$$c=\frac{uv}{{k}_{1}{k}_{2}}$$ 
Note. D’Alembert was an avid supporter (in fact, the coeditor) 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  20130322 14:36:06 
Last modified on  20130322 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 