# quadratic fields that are not isomorphic

Within this entry, $S$ denotes the set of all squarefree^{} integers not equal to $1$.

###### Theorem.

Let $m\mathrm{,}n\mathrm{\in}S$ with $m\mathrm{\ne}n$. Then $\mathrm{Q}\mathit{}\mathrm{(}\sqrt{m}\mathrm{)}$ and $\mathrm{Q}\mathit{}\mathrm{(}\sqrt{n}\mathrm{)}$ are not isomorphic^{} (http://planetmath.org/FieldIsomorphism).

###### Proof.

Suppose that $\mathbb{Q}(\sqrt{m})$ and $\mathbb{Q}(\sqrt{n})$ are isomorphic. Let $\phi :\mathbb{Q}(\sqrt{m})\to \mathbb{Q}(\sqrt{n})$ be a field isomorphism. Recall that field homomorphisms fix prime subfields. Thus, for every $x\in \mathbb{Q}$, $\phi (x)=x$.

Let $a,b\in \mathbb{Q}$ with $\phi (\sqrt{m})=a+b\sqrt{n}$. Since $\phi (a)=a$ and $\phi $ is injective^{}, $b\ne 0$. Also, $m=\phi (m)=\phi ({(\sqrt{m})}^{2})={(\phi (\sqrt{m}))}^{2}={(a+b\sqrt{n})}^{2}={a}^{2}+2ab\sqrt{n}+{b}^{2}n$. If $a\ne 0$, then $\sqrt{n}={\displaystyle \frac{m-{a}^{2}-{b}^{2}n}{2ab}}\in \mathbb{Q}$, a contradiction^{}. Thus, $a=0$. Therefore, $m={b}^{2}n$. Since $m$ is squarefree, ${b}^{2}=1$. Hence, $m=n$, a contradiction. It follows that $K$ and $L$ are not isomorphic.
∎

This yields an obvious corollary:

###### Corollary.

There are infinitely many distinct quadratic fields^{}.

###### Proof.

Note that there are infinitely many elements of $S$. Moreover, if $m$ and $n$ are distinct elements of $S$, then $\mathbb{Q}(\sqrt{m})$ and $\mathbb{Q}(\sqrt{n})$ 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 $\mathbb{Q}$ (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 |
---|---|

Canonical name | QuadraticFieldsThatAreNotIsomorphic |

Date of creation | 2013-03-22 16:19:44 |

Last modified on | 2013-03-22 16:19:44 |

Owner | Wkbj79 (1863) |

Last modified by | Wkbj79 (1863) |

Numerical id | 9 |

Author | Wkbj79 (1863) |

Entry type | Theorem |

Classification | msc 11R11 |