Theorem 1

(d'Alembert-Gauss). The field $\mathbf{C}$ of complex numbers is algebraically closed.

For the proof it is enough to establish that (i) every element $\geq 0$ in $\mathbf{R}$ has a square root, and (ii) every polynomial of odd degree with coefficients in $\mathbf{R}$ has at least one root in $\mathbf{R}$. The first of these assertions has already been proved in Chapter IV, § 3, no. 3. As to the second, if $f(X) = a_0 X^n + a_1 X^{n-1} + \cdots + a_n$ is a polynomial of odd degree $n$ ($a_0 \neq 0$) with real coefficients, we may write $f(x) = a_0 x^n g(x)$ for $x \neq 0$, where

tends to $+1$ as $x$ tends to $+\infty$ or $-\infty$. Hence there is a number $a > 0$ such that $f(a)$ has the sign of $a_0$ and $f(-a)$ the sign of $-a_0$;

and so by Bolzano’s theorem (Chapter IV, § 6, no. 1, Theorem 2), $f$ has at least one root in $[—a, a]$.