quadratic fields that are not isomorphic
Let with . Then and are not isomorphic (http://planetmath.org/FieldIsomorphism).
This yields an obvious corollary:
There are infinitely many distinct quadratic fields.
Note that there are infinitely many elements of . Moreover, if and are distinct elements of , then and are not isomorphic and thus cannot be equal. ∎
Note that the above corollary could have also been obtained by using the result regarding Galois groups of finite abelian extensions of (http://planetmath.org/GaloisGroupsOfFiniteAbelianExtensionsOfMathbbQ). On the other hand, using this result to prove the above corollary can be likened to “using a sledgehammer to kill a housefly”.
|Title||quadratic fields that are not isomorphic|
|Date of creation||2013-03-22 16:19:44|
|Last modified on||2013-03-22 16:19:44|
|Last modified by||Wkbj79 (1863)|