Quadratically closed field explained
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
for
n ≥ 0 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, 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.[4]
- A formally real Euclidean field E is not quadratically closed (as −1 is not a square in E) but the quadratic extension E is quadratically closed.[3]
- 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 Falg of F, as the union of all iterated quadratic extensions of F in Falg.[3]
Examples
- The quadratic closure of R is C.[3]
- The quadratic closure of
is the union of the
.
[3] - The quadratic closure of Q is the field of complex constructible numbers.
References
- Book: Lam, Tsit-Yuen . Tsit Yuen Lam
. Introduction to Quadratic Forms over Fields . 67 . . Tsit Yuen Lam . American Mathematical Society . 2005 . 0-8218-1095-2 . 1068.11023 . 2104929 .
- Book: Rajwade, A. R. . Squares . 171 . London Mathematical Society Lecture Note Series . . 1993 . 0-521-42668-5 . 0785.11022 .
Notes and References
- Lam (2005) p. 33
- Rajwade (1993) p. 230
- Lam (2005) p. 220
- Lam (2005) p. 34
- Lam (2005) p.270