The Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra is one of the most important theorems in mathematics, and is a result that every math student should know. It states that every non-constant polynomial with complex coefficients has at least one complex root.

Statement of the theorem

The statement of the theorem can be written as follows:

Every non-constant polynomial $p(z)$ of degree $n$ with complex coefficients has at least one complex root.

In other words, if you have a polynomial of the form

$p(z) = a_nz^n + a_{n-1}z^{n-1} + \cdots + a_1z + a_0$

where $a_n, a_{n-1}, \ldots, a_0$ are complex numbers and $a_n \neq 0$, then there exists a complex number $z_0$ such that $p(z_0) = 0$.

Historical background

The Fundamental Theorem of Algebra was first proved by Carl Friedrich Gauss in 1799, when he was just 21 years old. Gauss's proof was based on a clever argument involving factorization of polynomials, and it was quite different from the proofs that are commonly used today.

Over the years, many other mathematicians have contributed to the development of the theorem, and various different proofs have been discovered. Today, there are several different ways to prove the theorem, each with its own strengths and weaknesses.

A simple proof

One of the simplest proofs of the Fundamental Theorem of Algebra is based on the principle of mathematical induction. Here is an outline of the proof:

• Base case: If $n=1$, then $p(z) = a_1z + a_0$, which clearly has a complex root (namely, $z_0 = -a_0/a_1$).
• Induction step: Assume that the theorem is true for all polynomials of degree up to $n-1$. Now consider a polynomial $p(z)$ of degree $n$. If $p(z)$ has a complex root, then we are done. Otherwise, we can write $p(z)$ as $p(z) = (z-z_1)q(z)$, where $z_1$ is a complex number and $q(z)$ is a polynomial of degree $n-1$. (This follows from the Factor Theorem.) By the induction hypothesis, $q(z)$ has a complex root $z_2$. But then we have $p(z_2) = (z_2 - z_1)q(z_2) = 0$, so $z_2$ is a complex root of $p(z)$.

Conclusion

The Fundamental Theorem of Algebra is a fundamental result in mathematics that has many important applications. It is used in fields such as physics, engineering, computer science, and cryptography, and it is also a key tool in the study of complex analysis and algebraic geometry. Understanding this theorem and its proofs is essential for any student of mathematics, and it is a testament to the beauty and power of mathematical reasoning.

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