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 with one variable and coefficients that are complex numbers always has at least one solution, also known as a root. This means even if the numbers in the equation look very complicated, there is always a way to balance the equation.

This theorem works for equations with real numbers too, 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 collection of all complex numbers is a special mathematical world where every kind of these equations can be solved.

Another way to say this theorem is that any equation of a certain level of complexity, called degree n, will have exactly n solutions when we count them carefully, including when some solutions repeat. This idea connects many parts of mathematics and shows how the world of numbers fits together. Even though it has “fundamental” in its name, proving this theorem needs more advanced mathematics than basic algebra.

History

Peter Roth, in a book from 1608, said that equations can have as many answers as their highest power. Later, Albert Girard agreed but said the answers might not always be real numbers.

In the 1700s and 1800s, many smart people tried to fully prove this idea. Some got close but missed a few details. The first full proof was given by Argand in 1806. After that, others shared their own proofs and ideas to help us understand this important math rule better.

Equivalent statements

The Fundamental Theorem of Algebra can be described in several ways, but they all mean the same thing. One way says that any equation with numbers and letters (called a polynomial) that has more than one letter will always have at least one answer. This answer might use numbers that are not just regular whole numbers.

Another way to think about it is that any such equation can be split into smaller pieces. These smaller pieces are very simple and show us where the answers to the original equation are. This helps us see that answers always exist, even if we need special numbers to find them.

Proofs

To prove this important math idea, we need some ideas from mathematical analysis or topology, like the idea of continuity of functions. Some proofs also use ideas about differentiable or analytic functions.

Some proofs show that a polynomial with real numbers must have a complex root. This helps prove the bigger idea because any polynomial with complex numbers can be linked to one with real numbers.

There are many ways to prove this theorem. Some use ideas from complex analysis, like the maximum modulus principle or Liouville's theorem. Others use linear algebra and ideas about eigenvalues of matrices. There are also proofs that use topology, looking at how shapes change.

One way to think about it uses the idea of winding numbers. If a polynomial has no roots, it creates a special shape on a sphere. But this leads to a problem because the sphere isn’t “flat” in the way this shape needs it to be.

These different proofs show how math connects many areas, linking algebra, analysis, and geometry in interesting ways.

Corollaries

The fundamental theorem of algebra tells us that every polynomial equation has a solution. This means that complex numbers, which include all real numbers, are special and complete.

One important result is that any polynomial with real numbers as inputs can be broken down into simpler parts. These parts include equations with real solutions and equations that need imaginary numbers to solve them. This helps us understand how many solutions a polynomial might have.