Definition:Quadratic Irrational/Reduced/Associated

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Definition

Let $\alpha$ be a reduced quadratic irrational expressed as:

$\alpha = \dfrac{P + \sqrt D} Q$

Then $\alpha$ is associated to $D$ if and only if $Q$ divides $D - P^2$.


Example

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 $112$.


Also see