In mathematics, an Archimedean field is an ordered field with the Archimedean property. Ultimately, the eponym is the ancient Greek mathematician Archimedes of Syracuse.
In an ordered field F we can define the absolute value of an element x in F in the usual way by setting |x| = x for nonnegative x and |x| = −x for negative x. Then, an Archimedean field F is one such that for any x in F there exists n in the natural numbers N for which |x| < n.
The real numbers form an Archimedean field. Moreover, it can be proved that any archimedean field is isomorphic (as an ordered field) to a subfield of the real numbers.
Archimedean fields are important in the axiomatic construction of real numbers.
