Given a certain set of real numbers .  Suppose for any two real numbers in , , , and (if is non-zero) are also in , we say that is a field.  Addition, subtraction, multiplication and division are collectively called rational operations. Examples

• The set of all rational numbers is obviously a field.
• The set of all constructible numbers is also field because we have already shown that addition, subtraction, multiplication and division can be done by Euclidean constructions, which implies that if and are two constructible numbers, , , and (if is non-zero) are also constructible numbers.

A field is called Euclidean if for any positive real number  in , is also in . The set of all constructible numbers  is a Euclidean field because we can construct the square root of a length by Euclidean construction. Next, we are going to investigate the underlying structure of  in more detail.