\begin{definition}[Definition:Quadratic Irrational/Reduced]
An irrational root $\alpha$ of a quadratic equation with integer coefficients is a '''reduced quadratic irrational''' {{iff}}
:$(1): \quad \alpha > 1$
:$(2): \quad$ its conjugate $\tilde{\alpha}$ satisfies:
::::$-1 < \tilde{\alpha} < 0$
{{ExtractTheorem|Extract the following into separate pages}}
Solutions of such quadratics can be written as:
:$\alpha = \dfrac{\sqrt D + P} Q$
where $D, P, Q \in \Z$ and $D, Q > 0$.
It is also possible (though not required) to ensure that $Q$ divides $D - P^2$.
This is actually a necessary assumption for some proofs and warrants its own definition.
\end{definition}