Quadratic algebraic conjugate

If $\alpha$ is a quadratic irrational with $a\alpha^2+b\alpha+c=0$ , its quadratic algebraic conjugate is

\alpha'=-b/a-\alpha.

It is the other root of the same irreducible quadratic. Multiplying the polynomial by a nonzero rational constant does not change either root.

Product identity

The identities $\alpha+\alpha'=-b/a$ and $\alpha\alpha'=c/a$ imply

a(m+n\alpha)(m+n\alpha')=am^2-bmn+cn^2.

For integers $(m,n)\ne(0,0)$ , this is a nonzero integer. For example, the conjugate of $\sqrt2-1$ is $-\sqrt2-1$ . This algebraic conjugation differs from complex conjugation: both numbers here are real.

References

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