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

Examples

The quadratic closure of R is C.^[4]
The quadratic closure of F₅ is the union of the $F_{5^{2^{n}}}$ .^[4]
The quadratic closure of Q is the field of constructible numbers.

References

↑ Lam (2005) p. 33
1 2 Rajwade (1993) p. 230
1 2 Lam (2005) p. 34
1 2 3 4 5 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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