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2010 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

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]

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

1. Lam (2005) p. 33
2. Rajwade (1993) p. 230
3. Lam (2005) p. 34
4. Lam (2005) p. 220
5. Lam (2005) p.270
How to Cite This Entry:
Quadratically closed field. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Quadratically_closed_field&oldid=35484