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Pythagorean field (Definition)

Let $ F$ be a field. A field extension $ K$ of $ F$ is called a Pythagorean extension if $ K = F(\sqrt{1+\alpha^2})$ for some $ \alpha$ in $ F$, where $ \sqrt{1+\alpha^2}$ denotes a root of the polynomial $ x^2-(1+\alpha^2)$ in the algebraic closure $ \overline{F}$ of $ F$. A field $ F$ is Pythagorean if every Pythagorean extension of $ F$ is $ F$ itself.

The following are equivalent:

  1. $ F$ is Pythagorean
  2. Every sum of two squares in $ F$ is a square
  3. Every sum of (finite number of) squares in $ F$ is a square

Examples:

  • $ \mathbb{R}$ and $ \mathbb{C}$ are Pythagorean.
  • $ \mathbb{Q}$ 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 $ F_{py}$. Given a field $ F$, one way to construct its Pythagorean closure is as follows: let $ K$ be an extension over $ F$ such that there is a tower

$\displaystyle F=K_1\subseteq K_2\subseteq \cdots \subseteq K_n=K$
of fields with $ K_{i+1}=K_i(\sqrt{1+\alpha_i^2})$ for some $ \alpha_i\in K_i$, where $ i=1,\ldots,n-1$. Take the compositum $ L$ of the family $ \mathcal{K}$ of all such $ K$'s. Then $ L=F_{py}$.



"Pythagorean field" is owned by CWoo.
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Also defines:  Pythagorean extension, Pythagorean closure
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Cross-references: compositum, extension, contained, number, finite, sum, square, sum of two squares, the following are equivalent, algebraic closure, polynomial, root, field extension, field
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This is version 8 of Pythagorean field, born on 2004-05-21, modified 2007-06-07.
Object id is 5868, canonical name is PythagoreanField.
Accessed 3857 times total.

Classification:
AMS MSC: 12D15 (Field theory and polynomials :: Real and complex fields :: Fields related with sums of squares )
Pending Errata and Addenda
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