This site is supported by donations to The OEIS Foundation.

# Conjugates

Conjugates of an irreducible polynomial of degree ${\displaystyle n}$ appear in sets containing ${\displaystyle n}$ elements.

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

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

where ${\displaystyle D}$ is the discriminant of a quadratic polynomial.

### Properties

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

and

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

The case ${\displaystyle D=-1}$ gives complex conjugates.

${\displaystyle |\alpha |:={\sqrt {\alpha \cdot {\overline {\alpha }}}}={\sqrt {a^{2}-Db^{2}}}.\,}$
The case ${\displaystyle D=-1}$ gives the complex norm.