Fundamental Theorem of Algebra

Fundamental Theorem of Algebra: Let \( P(z) \) be a polynomial of degree \( n \) with complex coefficients. Then there exists at least one complex number \( z_0 \) such that \( P(z_0) = 0 \).

The Fundamental Theorem of Algebra is a cornerstone of mathematics and establishes a foundational result in the theory of polynomials.

It asserts that every polynomial of degree \( n \) with complex coefficients has exactly \( n \) roots in the complex numbers, counted according to their algebraic multiplicities.

Equivalently, any non-constant polynomial equation admits at least one solution in the complex plane.

The theorem was first proved by Carl Friedrich Gauss in 1799. His original proof, along with many later ones, relies on techniques from complex analysis, including the argument principle and Liouville's theorem.

This result is a central pillar of modern mathematics, as it guarantees the completeness of the complex numbers with respect to polynomial equations, ensuring that every complex polynomial of degree \( n \) has precisely \( n \) complex zeros.

To make the statement more explicit, consider a general polynomial of degree \( n \):

\[ P(z) = a_n z^n + a_{n-1} z^{n-1} + \cdots + a_1 z + a_0 \]

Here, \( a_n, a_{n-1}, \ldots, a_0 \) are complex coefficients, with \( a_n \neq 0 \).

The theorem guarantees the existence of at least one complex number \( z_0 \) such that \( P(z_0) = 0 \), and in fact ensures that the polynomial has exactly \( n \) such zeros when multiplicities are taken into account.

Practical example

Consider the following concrete example:

\[ P(z) = z^3 - 1 \]

The polynomial \( P(z) = z^3 - 1 \) has degree three and can be factored as:

\[ P(z) = (z - 1)(z^2 + z + 1) \]

From this factorization, it is immediate that \( z = 1 \) is a root. The remaining factor, \( z^2 + z + 1 \), can be solved using the quadratic formula:

\[ z = \frac{-1 \pm \sqrt{-3}}{2} = \frac{-1 \pm \sqrt{3}i}{2} \]

Thus, the roots of the polynomial are \( z = 1 \), \( z = \frac{-1 + \sqrt{3}i}{2} \), and \( z = \frac{-1 - \sqrt{3}i}{2} \). These three complex roots are exactly what the Fundamental Theorem of Algebra predicts.

This conclusion is not only of theoretical importance, but also has far-reaching practical implications across many areas of science and technology.

The Fundamental Theorem of Algebra not only guarantees the existence and number of roots of complex polynomials, but also underlies much of both pure and applied mathematics. It plays a key role in number theory by providing a structural framework for the study of polynomial equations, and it is essential in numerical analysis, where it supports the development and analysis of methods for approximating polynomial roots.