Quadratically closed field

2020 Mathematics Subject Classification: Primary: 12F05 [MSN][ZBL]

A field in which every element of the field has a square root in the field.[1][2]

## Contents

* 1 Examples
* 2 Properties
* 3 Quadratic closure
* 3.1 Examples
* 4 References

### Examples[edit]

* 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 $n \ge 0$ is quadratically closed but not algebraically closed.[3]
* The field of constructible numbers is quadratically closed but not algebraically closed.[4]

### Properties[edit]

* 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, $\mathbb{R}$ 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 $\mathbb{Z}$ 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 $E(\sqrt{-1})$ 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[edit]

A quadratic closure of a field $F$ is a quadratically closed field which embeds in any other quadratically closed field containing $F$. A quadratic closure for a given $F$ may be constructed as a subfield of the algebraic closure $F^{\mathrm{alg}}$ of $F$, as the union of all quadratic extensions of $F$ in $F^{\mathrm{alg}}$.[4]

### Examples[edit]

* The quadratic closure of the field of real numbers is the field of complex numbers.[4]
* The quadratic closure of the finite field $\mathbb{F}_5$ is the union of the $\mathbb{F}_{5^{2^n}}$.[4]
* The quadratic closure of the field of rational numbers is the field of constructible numbers.

## References[edit]

1. ↑ Lam (2005) p. 33
2. ↑ 2.0 2.1 Rajwade (1993) p. 230
3. ↑ 3.0 3.1 Lam (2005) p. 34
4. ↑ 4.0 4.1 4.2 4.3 4.4 Lam (2005) p. 220
5. ↑ Lam (2005) p.270

* Tsit Yuen Lam, Introduction to Quadratic Forms over Fields, Graduate Studies in Mathematics 67, American Mathematical Society (2005) ISBN 0-8218-1095-2 Zbl 1068.11023 MR2104929
* A. R. Rajwade, Squares, London Mathematical Society Lecture Note Series 171 Cambridge University Press (1993) ISBN 0-521-42668-5Zbl 0785.11022