Let be a field. A field extension of is called a Pythagorean extension if for some in , where denotes a root of the polynomial in the algebraic closure of . A field is Pythagorean if every Pythagorean extension of is itself.
The following are equivalent:
Every sum of two squares in is a square
Every sum of (finite number of) squares in 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 is called the Pythagorean closure of , and is written . Given a field , one way to construct its Pythagorean closure is as follows: let be an extension over such that there is a tower
of fields with for some , where . Take the compositum of the family of all such ’s. Then .
|Date of creation||2013-03-22 14:22:36|
|Last modified on||2013-03-22 14:22:36|
|Last modified by||CWoo (3771)|