# Quadratically closed field

In mathematics, a **quadratically closed field** is a field in which every element of the field has a square root in the field.[1][2]

## 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.[3] - The field of constructible numbers is quadratically closed but not algebraically closed.[4]

## 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.[2] - A field is quadratically closed if and only if its Witt–Grothendieck ring is isomorphic to
**Z**under the dimension mapping.[3] - 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.[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 closure *F*^{alg} of *F*, as the union of all quadratic extensions of *F* in *F*^{alg}.[4]

## References

- Lam (2005) p. 33
- Rajwade (1993) p. 230
- Lam (2005) p. 34
- Lam (2005) p. 220
- Lam (2005) p.270

- Lam, Tsit-Yuen (2005).
*Introduction to Quadratic Forms over Fields*. Graduate Studies in Mathematics.**67**. American Mathematical Society. ISBN 0-8218-1095-2. MR 2104929. Zbl 1068.11023. - Rajwade, A. R. (1993).
*Squares*. London Mathematical Society Lecture Note Series.**171**. Cambridge University Press. ISBN 0-521-42668-5. Zbl 0785.11022.

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