A **real closed field** is a type of field in which certain algebraic properties analogous to those of the real numbers hold. More formally, a field \( K \) is called a real closed field if it satisfies the following conditions: 1. **Algebraically Closed**: Every non-constant polynomial in one variable with coefficients in \( K \) has a root in \( K \).
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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.