C(X) über R(X)/Minimalpolynom/Aufgabe/Lösung

Es ist

${}{\begin{aligned}{\frac {X^{3}-{\left(2X^{4}-X\right)}{\mathrm {i} }}{X^{3}-1+X^{2}{\mathrm {i} }}}&={\frac {X^{3}-{\left(2X^{4}-X\right)}{\mathrm {i} }}{X^{3}-1+X^{2}{\mathrm {i} }}}\cdot {\frac {X^{3}-1-X^{2}{\mathrm {i} }}{X^{3}-1-X^{2}{\mathrm {i} }}}\\&={\frac {X^{3}{\left(X^{3}-1\right)}-{\left(2X^{4}-X\right)}X^{2}+{\left(-X^{5}-{\left(2X^{4}-X\right)}{\left(X^{3}-1\right)}\right)}{\mathrm {i} }}{{\left(X^{3}-1\right)}^{2}+X^{4}}}\\&={\frac {X^{6}-X^{3}-2X^{6}+X^{3}+{\left(-X^{5}-2X^{7}+2X^{4}+X^{4}-X\right)}{\mathrm {i} }}{X^{6}+X^{4}-2X^{3}+1}}\\&={\frac {-X^{6}+{\left(-2X^{7}-X^{5}+3X^{4}-X\right)}{\mathrm {i} }}{X^{6}+X^{4}-2X^{3}+1}}\\&={\frac {-X^{6}}{X^{6}+X^{4}-2X^{3}+1}}+{\frac {-2X^{7}-X^{5}+3X^{4}-X}{X^{6}+X^{4}-2X^{3}+1}}{\mathrm {i} }.\end{aligned}}$

Für ein Element ${}a+b{\mathrm {i} }\in K[{\mathrm {i} }]$ über einem Körper ${}K$ (in dem ${}-1$ kein Quadrat ist) ist das Minimalpolynom generell gleich

$T^{2}-2aT+a^{2}+b^{2}.$

Im vorliegenden Fall ist das Minimalpolynom also gleich

${}{\begin{aligned}&=T^{2}-2{\frac {-X^{6}}{X^{6}+X^{4}-2X^{3}+1}}T+{\left({\frac {-X^{6}}{X^{6}+X^{4}-2X^{3}+1}}\right)}^{2}+{\left({\frac {-2X^{7}-X^{5}+3X^{4}-X}{X^{6}+X^{4}-2X^{3}+1}}\right)}^{2}\\&=T^{2}+2{\frac {X^{6}}{X^{6}+X^{4}-2X^{3}+1}}T+{\frac {X^{12}+4X^{14}+X^{10}+9X^{8}+X^{2}+4X^{12}-12X^{11}+4X^{8}-6X^{9}+2X^{6}-6X^{5}}{X^{12}+X^{8}+4X^{6}+1+2X^{10}-4X^{9}+2X^{6}-4X^{7}+2X^{4}-4X^{3}}}\\&=T^{2}+2{\frac {X^{6}}{X^{6}+X^{4}-2X^{3}+1}}T+{\frac {4X^{14}+5X^{12}-12X^{11}+X^{10}-6X^{9}+13X^{8}+X^{2}+2X^{6}-6X^{5}}{X^{12}+2X^{10}-4X^{9}+X^{8}-4X^{7}+6X^{6}+2X^{4}-4X^{3}+1}}.\,\end{aligned}}$

Zur gelösten Aufgabe