Quadratically closed 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 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 ; 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 but the quadratic extension E 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.

  Quadratic closure

Examples

• The quadratic closure of R is C.
• The quadratic closure of F₅ is the union of the.
• The quadratic closure of Q is the field of constructible numbers.