In mathematics, a Euclidean field is an ordered field K for which every non-negative element is a square: that is, x ≥ 0 in K implies that x = y2 for some y in K. The constructible numbers form a Euclidean field.
It is the smallest Euclidean field, as every Euclidean field contains it as an ordered subfield.
In other words, the constructible numbers form the Euclidean closure of the rational numbers.
The following examples are also real closed fields.
The Euclidean closure of an ordered field K is an extension of K in the quadratic closure of K which is maximal with respect to being an ordered field with an order extending that of K.[5] It is also the smallest subfield of the algebraic closure of K that is a Euclidean field and is an ordered extension of K.