Algebraically closed field

In mathematics, an algebraically closed field is a special kind of number system where every non-constant polynomial can be solved. This means that for any equation you write using numbers from that field, there will always be a solution within the same field. For example, the real numbers are not algebraically closed because some equations, like x² + 1 = 0, have no real solutions. However, the complex numbers are algebraically closed, meaning every polynomial equation has at least one solution in the complex numbers.

Every field, such as the field of rational numbers, can be expanded to become algebraically closed. This expanded version is called an algebraic closure of the original field. Even if there are many ways to create such an expansion, they are all essentially the same, meaning they can be matched up in a way that keeps the original numbers in place.

Algebraically closed fields are important in many areas of mathematics. They show up in a chain of different types of number systems, from simple rngs all the way to these very complete fields, as shown in the list of class inclusions. These fields help mathematicians understand the solutions to equations and the structure of number systems.

Examples

The field of real numbers is not algebraically closed because the polynomial equation x² + 1 = 0 has no solution in real numbers, even though all its coefficients are real. By contrast, the fundamental theorem of algebra tells us that the field of complex numbers is algebraically closed, meaning every polynomial with complex number coefficients has a solution in complex numbers.

No finite field is algebraically closed. For example, if you list all the elements of a finite field, you can create a polynomial that has no solution within that field. However, when you combine all finite fields with the same characteristic, you get an algebraically closed field.

Equivalent properties

A field is called "algebraically closed" if every non-constant polynomial equation has a solution within that field. For example, the real numbers are not algebraically closed because the equation (x^2 + 1 = 0) has no real solutions. However, the complex numbers are algebraically closed because every polynomial equation has a solution there.

There are several ways to describe algebraically closed fields. One way is that every polynomial can be broken down into simpler pieces called "linear factors." Another way is that the field has no proper algebraic extensions, meaning it cannot be made bigger by adding new solutions to polynomial equations. These ideas all tie together to show what it means for a field to be algebraically closed.

Other properties

If a field is algebraically closed, it contains all the special numbers called _n_th roots of unity. These numbers are the solutions to the equation xⁿ − 1 = 0. This means that algebraically closed fields are also cyclotomically closed.

Every field can be extended to become algebraically closed. There is a special extension called the algebraic closure of the field. Additionally, the study of algebraically closed fields can be simplified using a method called quantifier elimination.