Fundamental theorem of algebra

The fundamental theorem of algebra, also called d'Alembert's theorem or the d'Alembert–Gauss theorem, tells us something important about equations. It says that every non-constant equation that has just one unknown number, and uses numbers called complex coefficients, will always have at least one solution called a root. This means even if the equation looks very complicated, there is always a number that makes it true!

This theorem also works for equations with real numbers because every real number can be thought of as a special kind of complex number, where the imaginary part is zero. In other words, the world of complex numbers is very complete — it cannot be made any bigger without changing what it already is. This is sometimes described by saying the field of complex numbers is algebraically closed.

Another way to say the theorem is that if you have an equation with a certain "degree" — like a quadratic (degree 2) or a cubic (degree 3) — it will have exactly that many solutions when you count them correctly, including how many times each solution appears, called its multiplicity. This idea comes from using a process called polynomial division.

History

Peter Roth wrote in 1608 that a math problem of degree n may have n answers. Albert Girard said in 1629 that such a problem has n answers, but he did not say they were real numbers.

Later, mathematicians tried to prove this idea. Leibniz made a mistake in 1702, and Nikolaus Bernoulli made a similar mistake. In 1742, a letter from Euler showed how to fix these mistakes.

Many famous mathematicians worked on proving this idea, including d'Alembert, Euler, de Foncenex, Lagrange, and Laplace. Their proofs had small errors or assumed things that were not proven.

The first complete proof was published by Argand in 1806. Later, Gauss also worked on proving the idea. The first book with a proof was written by Cauchy in 1821.

In the 1800s, mathematicians tried to find the answers directly. Weierstrass began this work, and later proofs were developed by others like Hellmuth Kneser and his son Martin Kneser.

Equivalent statements

The Fundamental Theorem of Algebra can be explained in a few different ways, and they all mean the same thing. One way says that any equation with more than one part, using real numbers, will always have at least one answer that includes imaginary numbers. Since real numbers can also be seen as imaginary numbers with zero imaginary part, this means these equations always have answers.

Another way to explain the theorem is that any such equation can be split into simpler pieces. For example, an equation of degree n can be shown as a product of n simpler equations, each with one answer. These answers are called the roots of the polynomial.

Proofs

All proofs of the Fundamental Theorem of Algebra use mathematical analysis. This includes the idea that functions can be continuous. Some proofs also use special types of functions.

One way to prove the theorem shows that any polynomial with real numbers as coefficients must have a solution in the complex numbers. This idea can also work for polynomials with complex coefficients. This works because complex numbers include all real numbers.

There are many different ways to prove this theorem. Some use ideas from complex analysis. Others use topology or algebra. All these methods show that every polynomial equation has at least one solution in the complex numbers.

Corollaries

The fundamental theorem of algebra tells us that every non-constant polynomial with complex numbers has at least one complex root. This means the complex numbers are "algebraically closed," so many important ideas in algebra work well with them.

Some key results that come from this theorem include:

• The complex numbers are the algebraic closure of the real numbers.
• Any polynomial with complex coefficients can be broken down into simpler parts.
• Polynomials with real coefficients can always be written using simpler pieces, and the number of non-real roots will always be even.