Find the quotient field of $\mathbb{Z}[\sqrt {2}]$

Question

My idea: how to make a connection between this with isomorphism.

Answer 1

$\mathbb Q[\sqrt 2]=\mathbb Q(\sqrt 2)$ because $\sqrt 2$ is algebraic.

$\mathbb Q(\sqrt 2)$ is the smallest subfield of $\mathbb C$ that contains $\mathbb Q$ and $\sqrt 2$.

$\mathbb Q(\sqrt 2)$ is thus the smallest subfield of $\mathbb C$ that contains $\mathbb Z$ and $\sqrt 2$.

Hence, $\mathbb Q(\sqrt 2)$ is the field of fractions of $\mathbb Z[\sqrt 2]$.

More generally, by the same argument, if $\alpha$ is an algebraic number, then $\mathbb Q[\alpha]=\mathbb Q(\alpha)$ and is the field of fractions of $\mathbb Z[\alpha]$.