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

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

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

Properties

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

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

and

$\alpha {\overline {\alpha }}=a^{2}-Db^{2}.\,$

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

Quadratic norm

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

$|\alpha |:={\sqrt {\alpha \cdot {\overline {\alpha }}}}={\sqrt {a^{2}-Db^{2}}}.\,$

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