Quadratically closed field

In mathematics, a quadratically closed field is a field of characteristic not equal to 2 in which every element has a square root. ^{[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 ${\displaystyle \mathbb {F} _{5^{2^{n}}}}$ 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 iterated quadratic extensions of F in F^alg. ^[4]

Examples

• The quadratic closure of R is C. ^[4]
• The quadratic closure of ${\displaystyle \mathbb {F} _{5}}$ is the union of the ${\displaystyle \mathbb {F} _{5^{2^{n}}}}$. ^[4]
• The quadratic closure of Q is the field of complex 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

• Lam, Tsit-Yuen (2005). Introduction to Quadratic Forms over Fields. Graduate Studies in Mathematics. Vol. 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. Vol. 171. Cambridge University Press. ISBN   0-521-42668-5. Zbl   0785.11022.