V.13 The Fundamental Theorem of Algebra

The COMPLEX NUMBERS [I.3 §1.5] can be thought of as what you obtain from the REAL NUMBERS [I.3 § 1.4] when you introduce a new number, denoted i, and stipulate that it is a solution of the equation x² = -1, or equivalently a root of the polynomial x² + 1. At first, this may seem an artificial thing to do—it is not obvious what is so important about x² + 1 as opposed to any other polynomial—but that is a judgment with which no professional mathematician would concur. The fundamental theorem of algebra is one of the best pieces of evidence that the complex number system is, in fact, natural, and natural in a profound way. It states that, within the complex number system, every polynomial has a root. In other words, once we introduce the number i, then not only can we solve the equation x² + 1 = 0, we can solve all polynomial equations (even if the coefficients are themselves complex). Thus, when one defines the complex numbers, one gets much more out of them than one puts in. It is this that makes them seem not an artificial construction but a wonderful discovery.

For many polynomials it is not hard to see that they have roots. For example, if P(x) = x^d – u for some positive integer d and some complex number u, then a root of P will be a dth root of u. One can write u in the form re^iθ, and then r^1/de^iθ/d will be such a root. This means that any polynomial that can be solved by a formula involving dth roots and the usual arithmetical operations, which includes all polynomials of degree less than 5, can be solved in the complex number system. However, owing to THE INSOLUBILITY OF THE QUINTIC [V.21], not all polynomials can be dealt with in this way, and in order to prove the fundamental theorem of algebra one must look for a less direct argument.

In fact, this is true even if one is looking for real roots of real polynomials. For example, if P(x) = 3x⁷ – 10x⁶ + x³ + 1, then we know that P(x) is large and positive when x is, since the x⁷ term is by far the most significant, and large and negative when x is, for the same reason. Therefore, at some point the graph of P crosses the x-axis, which means that there is some x with P(x) = 0. Notice that this argument does not tell us what x is—that is the sense in which it is “less direct.”

Now let us see how one might show that a polynomial has a complex root, by looking at the example P(x) = x⁴ + x² – 6x + 9. This can be rewritten x⁴ + (x – 3)², and since both x⁴ and (x - 3)² are nonnegative, and since they cannot be zero simultaneously, P cannot have a real root. To see that it has a complex root, we shall begin by fixing a large real number r and looking at the behavior of P(re^iθ) as θ varies between 0 and 2π. As θ varies in this way, re^iθ traces out a circle of radius r in the complex plane.

Now (re^iθ)⁴ = r⁴e^4iθ so the x⁴ part of P(re^iθ) traces out a circle of radius r⁴, but goes around it four times. If r is large enough, then the rest (that is, (re^iθ – 3)²) is so small compared with (re^iθ)⁴ that the only effect on the behavior of P(re^iθ) is to make it deviate very slightly from the circle of radius r⁴. This small deviation is not enough to stop the path of P(re^iθ) going around zero four times.

Next, let us consider what happens when r is very small. Then P(re^iθ) is very close to 9, whatever the value of θ, since (re^iθ)⁴, (re^iθ)², and (re^iθ) are all small. But this means that the path traced out by P(re^iθ) does not go around zero at all.

For any r we can ask how many times the path traced out by P(re^iθ) goes around zero. What we have just established is that for very large r the answer is four and for very small r it is zero. It follows that at some intermediate r the answer changes. But if you gradually shrink r, the path traced out by P(re^iθ) varies in a continuous way, so the only way this change can come about is if for some r the path crosses 0. This gives us the root we are looking for, since the path consists of points of the form P(re^iθ) and one of these points is 0.

Some care is needed to turn the above reasoning into a rigorous proof. However, this can be done, and it is not hard to generalize the resulting argument to one that applies to any polynomial.

The fundamental theorem of algebra is usually attributed to GAUSS [VI.26], who proved it in 1799 in his doctoral thesis. Though his argument (which was different from the one sketched above) was not fully rigorous by today’s standards, it was convincing and broadly correct. Later he went on to give three more proofs.