Definition:Quadratic Irrational/Reduced/Associated/Example

Example of Associated Quadratic Irrational
Consider the quadratic irrational $\alpha = \dfrac{2 + \sqrt 7} 4$.

While $\alpha$ is a reduced quadratic irrational, it is not associated to $7$.

However, if we write it as:
 * $\alpha = \dfrac {8 + \sqrt{112} } {16}$

the required condition holds.

Thus it is seen that $\alpha$ is associated to $112$.

Proof
Consider the conjugate $\tilde \alpha$ of $\alpha$:

and so $-1 < \tilde \alpha < 0$.

Thus $\alpha$ is a reduced quadratic irrational.

Note that:
 * $7 - 2^2 = 3$

and so $4 \nmid 7 - 2^2$.

So, by definition, $\alpha$ is not associated to $7$.

Now consider:

Now note that:
 * $112 - 8^2 = 48 = 4 \times 12$

and so $4 \mathrel \backslash \left({112 - 8^2}\right)$.

Thus, by definition, $\alpha$ is associated to $112$.