Euclidean field

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.

This article is about ordered fields. For algebraic number fields whose ring of integers has a Euclidean algorithm, see Norm-Euclidean field. For the class of models in statistical mechanics, see Euclidean field theory.

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.

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