2. Higher Degree Equations. The Fundamental Theorem of Algebra
The next breakthrough came in the 19th century, when Abel and Ruffini proved that equations of the fifth degree and above cannot be solved by radicals. Their results were considerably clarified by the work of Evariste Galois, who systematically developed the theory of fields.
Next, the work of Joseph Liouville on complex analysis made it possible to prove very simply that every complex valued polynomial has a root. This development is sketched below, without going into proofs of theorems in analysis.
Definition. Suppose f is a complex valued function defined in a neighbourhood U of a point a in the complex plane. f is said to be differentiable at a if there exists a complex number H such that, given any ε > 0, we can find a δ > 0, such that
| | | f(z) - f(a) | - H | | | < ε |
| | | | | |||
| | | z - a | | |
for all z with |z - a| < δ.
We then write
H = f'(a)
We say that a function f defined on U ⊆ C is analytic on U if it is differentiable at every point of U.
Liouville's Theorem. A function which is analytic and bounded in the whole plane must reduce to a constant.
[Proof on next page.]
The Fundamental Theorem of Algebra. If p is a complex valued polynomial of degree n ≥ 1, then p has at least one zero (point where p(z) = 0).
Proof. If p(z) = anzn + an-1zn-1 + ... + a1z + a0, we see that
p(z) = zn(an + an-1z-1 + ... + a1z-n + 1 + a0z-n)
∴ |p(z)| ≥ |z|n|an - |z|-1|an-1 + ... + a1z-n + 1 + a0z-n|
Since |z|-1 < 1 for |z| > 1, and |z|-1 → 0 as z → ∞, we see that there exists R > 0 such that
|z|-1|an-1 + ... + a1z-n + 1 + a0z-n| < |an|/2
for all z with |z| > R. For such z, we have
|p(z)| ≥ (|an|/2)|z|n
Assume that p(z) has no zero. The function f(z) = 1/p(z) is analytic on the whole plane, since the elementary rules of calculus hold for differentiable complex valued functions. For |z| > R we have
|f(z)| ≤ (2/|an|)|z|-n
which is evidently bounded. On the other hand, f(z) is also bounded on the compact region |z| ≤ R. Thus it is bounded in the entire plane and must reduce to a constant.
Thus we have proved by contradiction that p(z) has a zero.
Note. Liouville was also responsible for the publication of Galois's work in 1846, fourteen years after his death.