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.[2]
A formally real
Euclidean fieldE is not quadratically closed (as −1 is not a square in E) but the quadratic extension E(√−1) is quadratically closed.[4]
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.[5]
Quadratic closure
A quadratic closure of a field F is a quadratically closed field containing F which
embeds in any quadratically closed field containing F. A quadratic closure for any given F may be constructed as a subfield of the
algebraic closureFalg of F, as the union of all iterated quadratic extensions of F in Falg.[4]
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.[2]
A formally real
Euclidean fieldE is not quadratically closed (as −1 is not a square in E) but the quadratic extension E(√−1) is quadratically closed.[4]
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.[5]
Quadratic closure
A quadratic closure of a field F is a quadratically closed field containing F which
embeds in any quadratically closed field containing F. A quadratic closure for any given F may be constructed as a subfield of the
algebraic closureFalg of F, as the union of all iterated quadratic extensions of F in Falg.[4]