quadratic closure

A field K is said to be quadratically closed if it has no quadratic extensions. In other words, every element of K is a square. Two obvious examples are and 𝔽2.

A field K is said to be a quadratic closure of another field k if

  1. 1.

    K is quadratically closed, and

  2. 2.

    among all quadratically closed subfieldsMathworldPlanetmath of the algebraic closureMathworldPlanetmath k¯ of k, K is the smallest one.

By the second condition, a quadratic closure of a field is unique up to field isomorphism. So we say the quadratic closure of a field k, and we denote it by k~ Alternatively, the second condition on K can be replaced by the following:

K is the smallest field extension over k such that, if L is any field extension over k obtained by a finite number of quadratic extensions starting with k, then L is a subfield of K.


  • =~.

  • If 𝔼 is the Euclidean field in , then the quadratic extension 𝔼(-1) is the quadratic closure ~ of the rational numbersPlanetmathPlanetmathPlanetmath .

  • If k=𝔽5, consider the chain of fields


    Take the union of all these fields to obtain a field K. Then it can be shown that K=k~.

Title quadratic closure
Canonical name QuadraticClosure
Date of creation 2013-03-22 15:42:43
Last modified on 2013-03-22 15:42:43
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 8
Author CWoo (3771)
Entry type Definition
Classification msc 12F05
Defines quadratically closed