In mathematics, a quadratically closed field is a field in which every element of the field has a square root in the field.

## Examples

• The field of complex numbers is quadratically closed; more generally, any algebraically closed field is quadratically closed.
• The field of real numbers is not quadratically closed as it does not contain a square root of −1.
• The union of the finite fields $F_{5^{2^{n}}}$ for n  0 is quadratically closed but not algebraically closed.
• The field of constructible numbers is quadratically closed but not algebraically closed.

## Properties

• A field is quadratically closed if and only if it has universal invariant equal to 1.
• Every quadratically closed field is a Pythagorean field but not conversely (for example, R is Pythagorean); however, every non-formally real Pythagorean field is quadratically closed.
• A field is quadratically closed if and only if its Witt–Grothendieck ring is isomorphic to Z under the dimension mapping.
• A formally real Euclidean field E is not quadratically closed (as −1 is not a square in E) but the quadratic extension E(−1) is quadratically closed.
• Let E/F be a finite extension where E is quadratically closed. Either −1 is a square in F and F is quadratically closed, or −1 is not a square in F and F is Euclidean. This "going-down theorem" may be deduced from the Diller–Dress theorem.

• The quadratic closure of F5 is the union of the $F_{5^{2^{n}}}$ .