Quadratic irrational

A quadratic irrational is an irrational real number $\alpha$ satisfying

a\alpha^2+b\alpha+c=0

for some integers $a,b,c$ with $a\ne0$ . Because $\alpha$ is irrational, the polynomial cannot factor into linear polynomials over $\mathbb Q$ . Its discriminant $b^2-4ac$ is positive and is not an integer square.

Examples

The numbers $\sqrt2$ and $\sqrt2-1$ satisfy $x^2-2=0$ and $x^2+2x-1=0$ , respectively. Each has a second real root, its quadratic algebraic conjugate. The two roots together provide arithmetic lower bounds on nonzero integer linear forms.

References

Fukshansky, Diophantine Approximation, §3 (quadratic irrationals).