Real closed field

In mathematics, a real closed field is a special kind of field that shares the same basic properties as the field of real numbers. This means that many important mathematical ideas and rules that work for real numbers can also be used in these fields. Real closed fields are important because they let mathematicians study and solve problems in a broader setting.

One key feature of a real closed field is that it follows the same first-order properties as the real numbers. These properties can be described using logic symbols and arithmetic operations, without needing to refer to more complex structures like sequences or functions. This makes real closed fields a useful tool in algebra and other areas of mathematics.

Examples of real closed fields include the familiar field of real numbers, the field of real algebraic numbers, and even more complex fields like the hyperreal numbers, which contain infinitesimally small numbers. By studying real closed fields, mathematicians can often prove general versions of theorems that originally were about the real numbers alone.

Equivalent definitions

A real closed field is a special kind of number system that behaves very much like the real numbers we use every day. One way to think about it is that it has the same basic rules and properties as the real numbers when we look at statements we can express with simple math logic.

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

Examples of real closed fields

Some fields behave just like the real numbers in important ways. 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 explains that any ordered field, like the field of rational numbers, can be extended to become a real closed field. This extended field, called the real closure, keeps the original ordering and is unique in its properties.

For example, the real closure of the rational numbers is the field of real algebraic numbers. Even if a field isn’t ordered, it can still have a real closure, which might be a ring instead of a field. This idea helps mathematicians understand the properties of numbers and their extensions. 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 includes symbols for addition, multiplication, the numbers 0 and 1, and the order relation ≤. In this language, the theory of real closed fields consists of all sentences that follow from certain basic rules. These rules include the properties of ordered fields, the idea that every positive number has a square root, and that certain types of equations always have solutions.

A mathematician named Tarski showed that this theory is complete and decidable. This means that there is a way to determine whether any statement about real closed fields is true or false. This was achieved by a method called quantifier elimination, which simplifies statements so they can be easily checked. These important results were discovered around 1930 and published in 1948.

Order properties

A real closed field can be organized so that we know which numbers are positive — these are exactly the squares of non-zero numbers. An important feature of the real numbers is that for any real number, we can find a whole number that is larger. However, there are other real closed fields that are not like this; for example, fields that include very large numbers or very small numbers that are close to zero. These special fields help mathematicians understand more about the structure of numbers and how they relate 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 assumption, all real closed fields with the same size and a certain property called η₁ are essentially the same. One way to build such a field is by using something called an ultrapower, which involves the real numbers and special mathematical structures.

Another way to create this field is by using series of real numbers with only countably many non-zero terms. This field, while not complete on its own, can be expanded to a larger field that is useful in a branch of mathematics called nonstandard analysis. This larger field has more elements than the original and includes it as a part of itself.

Elementary Euclidean geometry

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