A formally real field is a type of field in mathematics that adheres to certain properties regarding sums of squares. Specifically, a field \( K \) is said to be formally real if it does not contain any non-negative elements that cannot be expressed as a sum of squares of elements from \( K \).
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A field is formally real when is not a finite sum of squares. The Artin-Schreier ordering criterion says this is equivalent to the existence of an ordering compatible with the field operations. Such a field has characteristic zero and embeds into a real closed field.