A field is said to be a quadratic closure of another field if
is quadratically closed, and
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 .
If , consider the chain of fields
Take the union of all these fields to obtain a field . Then it can be shown that .
|Date of creation||2013-03-22 15:42:43|
|Last modified on||2013-03-22 15:42:43|
|Last modified by||CWoo (3771)|