Quadratically closed field
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In mathematics, a quadratically closed field is a field 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 [math]\displaystyle{ \mathbb F_{5^{2^n}} }[/math] 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 Falg of F, as the union of all iterated quadratic extensions of F in Falg.[4]
Examples
- The quadratic closure of R is C.[4]
- The quadratic closure of [math]\displaystyle{ \mathbb F_5 }[/math] is the union of the [math]\displaystyle{ \mathbb F_{5^{2^n}} }[/math].[4]
- The quadratic closure of Q is the field of complex constructible numbers.
References
- Lam, Tsit-Yuen (2005). Introduction to Quadratic Forms over Fields. Graduate Studies in Mathematics. 67. American Mathematical Society. ISBN 0-8218-1095-2.
- Rajwade, A. R. (1993). Squares. London Mathematical Society Lecture Note Series. 171. Cambridge University Press. ISBN 0-521-42668-5.
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