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# quadratic fields that are not isomorphic

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

###### Theorem.

Let $m,n\in S$ with $m\neq n$. Then $\mathbb{Q}(\sqrt{m})$ and $\mathbb{Q}(\sqrt{n})$ are not isomorphic.

###### Proof.

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

Let $a,b\in\mathbb{Q}$ with $\varphi(\sqrt{m})=a+b\sqrt{n}$. Since $\varphi(a)=a$ and $\varphi$ is injective, $b\neq 0$. Also, $m=\varphi(m)=\varphi((\sqrt{m})^{2})=(\varphi(\sqrt{m}))^{2}=(a+b\sqrt{n})^{2}% =a^{2}+2ab\sqrt{n}+b^{2}n$. If $a\neq 0$, then $\displaystyle\sqrt{n}=\frac{m-a^{2}-b^{2}n}{2ab}\in\mathbb{Q}$, 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. ∎

###### 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 $\mathbb{Q}$. On the other hand, using this result to prove the above corollary can be likened to “using a sledgehammer to kill a housefly”.

## Mathematics Subject Classification

11R11*no label found*

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