DG

Fundamental theorem of algebra

Every complex polynomial factors into constant and linear factors.

Splitting one root: If z1 is a root of a polynomial p(z), then z-z1 divides p(z).
(Harriot, 1560-1621, Descartes, La Géométrie, 1637)

    This can be easily seen from long division of polynomials.
    There exist polynomials q and r such that p(z)=(z-z1)q(z)+r(z), where the deg r < 1.
    Therefore, r is a constant. Plugging in z1 for z we obtain r=0, so p(z)=(z-z1)q(z).
Total factorization: If q has a root z2, split z-z2 off q, as above, and proceed by reverse induction on degree (each successive factor has degree 1 less than the preceding) until a polynomial factor without roots remains.

Thus, it suffices to show that a polynomial without roots must be constant.

Suppose p(z)=a0+a1z+...+anzn is a polynomial of degree n > 0.
If a0=0, then z=0 is a root, so assume that a0 is not zero.

Images of circles:

    Take a circle centered at the origin of radius r.
    The image of this circle under p is a closed curve in the plane.
    The following graph (generated with Maple) shows such curves for various values of r for a particular polynomial.

Index of a curve: If any of the curves (for various r) hits the origin, we have a root and are done.
For any closed curve missing the origin we can define the winding number w as the number of times the curve goes around the origin.

For small r (e.g. the black circle above) p(z) hovers around a0, so w=0.

For large r (e.g. the blue circle above) p(z) is close to anzn, so w=n.

If none of the curves meets the origin, the winding number w(r) is defined for all r in the interval [0, \infty). By continuity of w with respect to r the image of w is a connected subset of Z, so a point. In other words, the winding number does not depend on r, so n=0, so p is a constant. QED

In the example above, you can see how with increasing r the curve gets bigger (black -> cyan -> magenta -> blue) and crosses the origin along the way. Note how the magenta curve crosses the origin twice, corresponding to the two imaginary roots of the same modulus. The blue curve goes around the origin 3 times, corresponding to the degree of the polynomial.

Argument principle: If we start with a simple positively oriented loop (above we used a circle), then the winding number of its image around the origin w is equal to the number of zeros of the function, counted with multiplicity, inside the original loop.

Last updated: Jun 17 2008 / Last fetched: Sun Nov 22 21:22:14 CST 2009