Pythagorean field
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:
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1.
is Pythagorean
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2.
Every sum of two squares in is a square
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3.
Every sum of (finite number of) squares in is a square
Examples:
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•
and are Pythagorean.
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•
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 .
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 |