Quadratically closed field

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 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 F5 is the union of the .[4]
  • The quadratic closure of Q is the field of constructible numbers.
gollark: Oh, go consume bees.
gollark: Anyway, potatOS now includes a high-performance* factorization engine using the Pollard rho algorithm, [DATA EXPUNGED], and the Rabin-Miller primality test.
gollark: How do you know about PotatOS Lightcycle? It's classified.
gollark: potato operation system.
gollark: Loading self-replicator beeoids... makes it unusably slow, great.

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. 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.
This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.