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quadratic closure
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(Definition)
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A field  is said to be quadratically closed if it has no quadratic extensions. In other words, every element of  is a square. Two obvious examples are
 and
 .
A field  is said to be a quadratic closure of another field  if
 is quadratically closed, and- among all quadratically closed subfields of the algebraic closure
 of  ,  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  , and we denote it by
 Alternatively, the second condition on  can be replaced by the following:
 is the smallest field extension over  such that, if  is any field extension over  obtained by a finite number of quadratic extensions starting with  , then  is a subfield
of  .
Examples.
-
 .
- If
 is the Euclidean field in
 , then the quadratic extension
 is the quadratic closure
 of the rational numbers
 .
- If
 , consider the chain of fields
Take the union of all these fields to obtain a field  . Then it can be shown that
 .
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"quadratic closure" is owned by CWoo.
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(view preamble)
| Also defines: |
quadratically closed |
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Cross-references: union, chain, rational numbers, Euclidean field, number, field extension, field isomorphism, algebraic closure, subfields, obvious, square, quadratic extensions, field
There is 1 reference to this entry.
This is version 5 of quadratic closure, born on 2006-02-27, modified 2006-02-28.
Object id is 7659, canonical name is QuadraticClosure.
Accessed 1241 times total.
Classification:
| AMS MSC: | 12F05 (Field theory and polynomials :: Field extensions :: Algebraic extensions) |
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Pending Errata and Addenda
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![$\displaystyle k\le k(\sqrt{2})\le k(\sqrt[4]{2})\le \cdots \le k(\sqrt[2^n]{2})\le \cdots $](http://images.planetmath.org:8080/cache/objects/7659/l2h/img28.png "$\displaystyle k\le k(\sqrt{2})\le k(\sqrt[4]{2})\le \cdots \le k(\sqrt[2^n]{2})\le \cdots $"){kind=small}