Fundamental theorem of algebra

The fundamental theorem of algebra, also called d'Alembert's theorem or the d'Alembert–Gauss theorem, tells us something very 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. Even though it has "fundamental" in its name, this theorem was named back when "algebra" mostly meant solving equations, and proving it needs more advanced mathematics than basic algebra.

History

Peter Roth, in his book Arithmetica Philosophica published in 1608, wrote that a polynomial equation of degree n may have n solutions. Albert Girard, in his book L'invention nouvelle en l'Algèbre published in 1629, said that a polynomial equation of degree n has n solutions, though he did not say they had to be 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 correct these mistakes.

Many famous mathematicians worked on proving this theorem over the years, including d'Alembert, Euler, de Foncenex, Lagrange, and Laplace. Their proofs had gaps or made assumptions that were not fully proven.

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

In the 1800s, mathematicians looked for ways to find the solutions directly, not just prove they exist. Weierstrass started 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 described in several different but equivalent ways. One way says that any equation with numbers and letters (called a polynomial) that has more than one term and uses real numbers will always have at least one solution that includes imaginary numbers. Since all real numbers can also be thought of as imaginary numbers with zero imaginary part, this means that these equations always have solutions.

Another way to describe the theorem is that any such equation can be broken down into simpler pieces. For example, an equation of degree n can be written as a product of n simpler equations, each of which has one solution. These solutions are called the roots of the polynomial.

Proofs

All proofs of the Fundamental Theorem of Algebra involve mathematical analysis, particularly the concept of continuity of functions. Some proofs also use differentiable or analytic functions. Because of this, the theorem is sometimes described as not being truly fundamental to algebra.

One approach to proving the theorem shows that any non-constant polynomial with real coefficients must have a complex root. This can then be extended to polynomials with complex coefficients. This method works because complex numbers include all real numbers.

There are many different types of proofs for this theorem. Some use ideas from complex analysis, while others use topology or algebra. Despite the different methods, they all confirm that every non-constant 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 apply to 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.