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 $112$.