Fundamental Theorem of Algebra

Every Polynomial equation having Complex Coefficients and degree $\geq 1$ has at least one Complex Root. This theorem was first proven by Gauß. It is equivalent to the statement that a Polynomial $P(z)$ of degree $n$ has $n$ values of $z$ (some of them possibly degenerate) for which $P(z) = 0$. An example of a Polynomial with a single Root of multiplicity $>1$ is $z^2-2z+1=(z-1)(z-1)$, which has $z=1$ as a Root of multiplicity 2.

See also Degenerate, Polynomial

References

Courant, R. and Robbins, H. ``The Fundamental Theorem of Algebra.'' §2.5.4 in What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 101-103, 1996.