Quadratically closed field
In mathematics, a quadratically closed field is a field of characteristic not equal to 2 in which every element has a square root.
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.
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
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 closure Falg of F, as the union of all iterated quadratic extensions of F in Falg.Examples
- The quadratic closure of R is C.
- The quadratic closure of is the union of the.
- The quadratic closure of Q is the field of complex constructible numbers.