Pythagorean field

Let F be a field. A field extension K of F is called a Pythagorean extensionMathworldPlanetmath if K=F(1+α2) for some α in F, where 1+α2 denotes a root of the polynomialPlanetmathPlanetmath x2-(1+α2) in the algebraic closureMathworldPlanetmath F¯ of F. A field F is Pythagorean if every Pythagorean extension of F is F itself.

The following are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath:

  1. 1.

    F is Pythagorean

  2. 2.

    Every sum of two squares in F is a square

  3. 3.

    Every sum of (finite number of) squares in F is a square


  • and are Pythagorean.

  • is not Pythagorean.

Remark. Every field is contained in some Pythagorean field. The smallest Pythagorean field over a field F is called the Pythagorean closure of F, and is written Fpy. Given a field F, one way to construct its Pythagorean closure is as follows: let K be an extensionPlanetmathPlanetmath over F such that there is a tower


of fields with Ki+1=Ki(1+αi2) for some αiKi, where i=1,,n-1. Take the compositum L of the family 𝒦 of all such K’s. Then L=Fpy.

Title Pythagorean field
Canonical name PythagoreanField
Date of creation 2013-03-22 14:22:36
Last modified on 2013-03-22 14:22:36
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 11
Author CWoo (3771)
Entry type Definition
Classification msc 12D15
Defines Pythagorean extension
Defines Pythagorean closure