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.

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.
• The field of constructible numbers 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 (for example, $\mathbb{R}$ is Pythagorean); 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 $\mathbb{Z}$ under the dimension mapping.
• 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.
• 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 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 $F^{\mathrm{alg}}$.

Examples

• The quadratic closure of the field of real numbers is the field of complex numbers.
• The quadratic closure of the finite field $\mathbb{F}_5$ is the union of the $\mathbb{F}_{5^{2^n}}$.
• 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 0-521-42668-5 Zbl 0785.11022