Conjugates

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Conjugates of an irreducible polynomial of degree $n$ appear in sets containing $n$ elements.

Quadratic conjugates

Quadratic conjugates are pairs of quadratic numbers $a+b\sqrt{D}$ and $a-b\sqrt{D}$. The quadratic conjugate of a quadratic number $\alpha =a+b\sqrt{D}$ is defined as

$\bar{\alpha }:=a-b\sqrt{D},$

where $D$ is the discriminant of a quadratic polynomial.

Properties

$\frac{\alpha +\bar{\alpha }}{2}=\frac{(a+b\sqrt{D})+(a-b\sqrt{D})}{2}=a,$

$\frac{\alpha -\bar{\alpha }}{2\sqrt{D}}=\frac{(a+b\sqrt{D})-(a-b\sqrt{D})}{2\sqrt{D}}=b,$

and

$\alpha \bar{\alpha }=a^2-D{b}^2.$

The case $D=-1$ gives complex conjugates.

Quadratic norm

The quadratic norm (i.e. the norm of a quadratic number) is defined as

$|\alpha |:=\sqrt{\alpha \cdot \bar{\alpha }}=\sqrt{a^2-D{b}^2}.$

The case $D=-1$ gives the complex norm.