Fundamental theorem of algebra

Every complex polynomial (including real polynomials) has at least one complex root. Actually, each complex polynomial of degree n has n roots, but some of those roots may be the same.

A more general concept is algebraic closure. A field F is algebraically closed if every nonconstant polynomial over F splits over F. The FTA says that C is algebraically closed. For more information on polynomials, splitting fields, and algebraic closure, consult your favourite algebra book's chapter on field extensions. For a proof of the FTA, consult any book on complex analysis---the proof is not very complicated, but it relies on more than just algebra.

As an illustration, it should be pointed out that this theorem is equivalent to the statement that every polynomial

C_nxⁿ + C_n-1x^n-1 + ... + C₁x + C₀

can be factored, that is put in a form

(x-r₁) * (x-r₂) * ... * (x-r_n)

which, when expanded algebraically and simplified, will yield the original polynomial.

r₁, r₂, etc., are the roots of the polynomial. The theorem doesn't show how to factor the polynomial; indeed, there is no general algorithm for factoring polynomials, although Laguerre's Method comes close.

There's a nice proof of this which uses Liouville's theorem. For convenience I'll restate it here:

Theorem (Liouville) Let f be holomorphic and bounded on the entire complex plane. Then f is constant.

Theorem (Fundamental theorem of algebra) Let p be a non-constant polynomial with complex coefficients. Then p has a root, ie. there is a complex number z with p(z) = 0.

Proof Suppose there is no such number z. We show that p is constant.

Let f = 1/p. Then f is holomorphic wherever p is non-zero, which by assumption is the whole complex plane.

Moreover, p is a polynomial, so |p(w)| tends to infinity as |w| tends to infinity; consequently, |f(w)| tends to 0 as |w| tends to infinity. So there exists M such that |f(w)| < 1 whenever |w| > M. And f is holomorphic, and hence bounded, on the compact set {w : |w| <= M}. So f is bounded on the complex plane.

So f satisfies the conditions for Liouville's theorem, and must be constant. Hence p is constant. QED

The Fundamental Theorem of Algebra (unlike that of calculus) is usually implicitly understood since middle school, but not often stated in explicit terms. Simply stated, every polynomial with real coefficients factors into linear and quadratic terms, or has certain real and certain conjugate pairs of roots.

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