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A field can be ordered compatibly with addition and multiplication if and only if it is a formally real field. This ordering criterion concerns real algebra, and is different from Artin–Schreier theory of cyclic extensions in positive characteristic.

A formally real field is a field in which $-1$ is not a finite sum of squares. The theory of formally real fields consists of the field axioms and, for each $m\ge 1$, the sentenceA real closed field is a formally real field with no proper formally real algebraic extension. An equivalent first-order axiomatization of the Theory of real closed fields adds: every $a$ or its negative is a square, and every odd-degree polynomial has a root. Explicitly the square axiom isand the polynomial axioms say that every monic polynomial of degree $2d+1$ has a root, for each $d\ge 0$.

The Artin-Schreier ordering criterion says that every formally real field can be ordered, and the real closure theorem embeds each ordered field into an algebraic real closed field extension. Consequently every FRF model embeds into an RCF model. Conversely every RCF model is itself an FRF model, so the reverse embedding requirement is automatic.

It remains to establish model completeness. A real closed field has a unique order, definable in the field language byAny field embedding between real closed fields preserves this order: positive elements are nonzero squares, and negative elements have positive negatives. The quantifier elimination for ordered real closed fields theorem makes every such ordered embedding elementary, by the preceding part. Restricting to formulas of the field language makes the original embedding elementary as well. Thus RCF is model-complete in the field language.

The embedding conditions and model completeness proveThe quantifier elimination invoked here is in the ordered language. In the unordered field language, model completeness still holds, but full quantifier elimination does not follow from forgetting the order.

A real closed field is a formally real field with no proper formally real algebraic extension. Equivalently every positive element in its unique compatible order is a square and every odd-degree polynomial has a root. Its positive cone is definable in the field language as the nonzero squares. The Theory of real closed fields is model-complete.

Every ordered field has an ordered algebraic real closure. Combined with the Artin-Schreier ordering criterion, every formally real field embeds into a real closed field. This is one embedding direction in the model companion relation between their theories.