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

F
2n
5
for n ≥ 0 is quadratically closed but not algebraically closed.

Properties

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

F5

is the union of the
F
2n
5
.[3]

References

. Introduction to Quadratic Forms over Fields . 67 . . Tsit Yuen Lam . American Mathematical Society . 2005 . 0-8218-1095-2 . 1068.11023 . 2104929 .

Notes and References

  1. Lam (2005) p. 33
  2. Rajwade (1993) p. 230
  3. Lam (2005) p. 220
  4. Lam (2005) p. 34
  5. Lam (2005) p.270