Real-closed field

Definition

A real-closed field is a field satisfying the following equivalent conditions:

1. It is elementarily equivalent to the field of real numbers.
2. It is a Euclidean field (i.e., it is formally real and for every element $D$, $D$ or $-D$ is a square) and every polynomial of odd degree over the field has a root.
3. Its algebraic closure is a degree two extension of it, given as the splitting field of the polynomial $x^{2}+1$.

Examples

• Field of real numbers
• Field of real algebraic numbers

Relation with other properties

Weaker properties

• Euclidean field
• Pythagorean field
• Formally real field
• Field with trivial automorphism group