# Quadratically closed field

From Encyclopedia of Mathematics

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]}

### 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 $F_{5^{2^n}}$ 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

- 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

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

- 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

- 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

**How to Cite This Entry:**

Quadratically closed field.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Quadratically_closed_field&oldid=54691