Quadratic number fields

Let $a$ be any integer which is not the perfect square of another integer. Let us abbreviate $\alpha=\sqrt{a}$ and define

$${\mathbb{Q}}(\alpha)=\{x+\alpha y\ \vert\ x,y\in {\mathbb{Q}}\}.$$

This is called the quadratic extension of the rationals associated to $\alpha$. It is an extension since it contains the field ${\mathbb{Q}}$ as a subset. It is called an imaginary quadratic field if $a<0$ and a real quadratic field if $a>0$.

For brevity, let $K={\mathbb{Q}}(\alpha)$. Multiplication in $K$ is given by

$$(x+\alpha y)(x+\alpha y)=xx'+ayy'+\alpha (xy'+x'+y),$$

for $x,x',y,y'\in {\mathbb{Q}}$. Inverse is given by

$$(x+\alpha y)^{-1}={x-\alpha y\over x^2+a y^2}.$$

Since $a$ is not a perfect square, $x^2+a y^2\not=0$ for rational $x,y$ unless $x=y=0$.