Real closed field

In mathematics, a real closed field is a special kind of field. It has the same basic properties as the field of real numbers. This means many math ideas that work for real numbers can also work in these fields.

Real closed fields are important. They help mathematicians study and solve problems in many areas.

One key feature of a real closed field is that it follows the same basic rules as the real numbers. These rules can be described using logic symbols and arithmetic, without needing more complex ideas.

Examples of real closed fields include the real numbers, the field of real algebraic numbers, and even more complex fields like the hyperreal numbers. Studying real closed fields helps mathematicians understand theorems about real numbers better.

Equivalent definitions

A real closed field is a special kind of number system that works a lot like the real numbers we use every day. It has the same basic rules and properties as the real numbers when we look at simple math statements.

Some examples of real closed fields include the real numbers themselves, the real numbers that can be written using basic operations and roots, and some number systems that include very small or very large numbers. This idea helps mathematicians because many important facts about real numbers still work in any real closed field.

Examples of real closed fields

Some fields act a lot like the real numbers. These special fields are called real closed fields. Examples include the field of real numbers itself, the field of real algebraic numbers, and the hyperreal number fields. Other examples are the field of computable real numbers, the field of definable real numbers, the field of Puiseux series with real coefficients, the Levi-Civita field, the superreal number fields, and the field of surreal numbers.

Real closure

The Artin–Schreier theorem shows that any ordered field, like the field of rational numbers, can be made into a real closed field. This new field, called the real closure, keeps the same order as the original field and has special properties.

For example, the real closure of the rational numbers is the field of real algebraic numbers. Even if a field does not have a natural order, it can still have a real closure, which might be a ring instead of a field. This idea helps mathematicians learn more about numbers and how they can be extended. The theorem is named after Emil Artin and Otto Schreier, who proved it in 1926.

Decidability and quantifier elimination

The language of real closed fields has symbols for addition, multiplication, and numbers like 0 and 1. It also includes a way to show if one number is bigger or smaller than another.

Mathematicians use some basic rules to understand real closed fields. These rules talk about how numbers work together, how every positive number has a square root, and how some equations always have answers.

A mathematician named Tarski discovered something important. He showed that we can always decide if a statement about real closed fields is true or false. He used a method called quantifier elimination to make statements simpler and easier to check. Tarski found these results around 1930 and shared them in 1948.

Order properties

A real closed field can be set up so we can tell which numbers are positive. These are the squares of numbers that are not zero. One key part of the real numbers is that for any real number, we can find a whole number that is bigger. But there are other real closed fields that are not like this. For example, some fields have very big numbers or very small numbers close to zero. These special fields help mathematicians learn more about how numbers are built and how they connect to each other.

The generalized continuum hypothesis

If we assume the generalized continuum hypothesis, the properties of real closed fields become much simpler. Under this idea, all real closed fields that have the same size and a special property called η₁ are basically the same.

One way to build such a field is by using something called an ultrapower. This uses the real numbers and some special math structures.

Another way to create this field is by using series of real numbers that have only countably many non-zero terms. This field is not complete by itself, but it can be made bigger. The bigger field is useful in a part of math called nonstandard analysis. The bigger field has more elements than the original and includes it as a part of itself.

Elementary Euclidean geometry

Tarski's axioms are rules that explain the basic ideas of Euclidean geometry. These axioms show us that points on a straight line can be treated like numbers in a special number system called a real closed field. With these numbers, we can place points on a flat surface, making a coordinate system that looks like a flat plane. Because of these properties, mathematicians can answer many questions about Euclidean geometry using logic and decisions based on these rules.