# Fünfter Kreisteilungsring/Gitter/Ganzheitsbasis der Potenzen/Reelle Ganzheitsmatrix/Aufgabe/Lösung

Es ist

${\displaystyle {}R_{5}=\mathbb {Z} [X]/{\left(X^{4}+X^{3}+X^{2}+X+1\right)}\,,}$

die vier komplexen Einbettungen sind durch ${\displaystyle {}X\mapsto \zeta ,\zeta ^{2},\zeta ^{3},\zeta ^{4}}$ gegeben, wobei die beiden äußeren und die beiden inneren zueinander konjugiert sind. Die beiden für die reelle Gesamteinbettung relevanten Einbettung sind also

${\displaystyle \varphi \colon R_{5}\longrightarrow {\mathbb {C} }\cong \mathbb {R} ^{2},\,X\longmapsto \zeta ,}$

und

${\displaystyle \psi \colon R_{5}\longrightarrow {\mathbb {C} }\cong \mathbb {R} ^{2},\,X\longmapsto \zeta ^{2}.}$

Dabei wird die Ganzheitsbasis auf

{\displaystyle {}{\begin{aligned}\left(\varphi (X),\,\varphi (X^{2}),\,\varphi (X^{3}),\,\varphi (X^{4})\right)&=\left(\zeta ,\,\zeta ^{2},\,\zeta ^{3},\,\zeta ^{4}\right)\\&=\left(\cos {\frac {2\pi }{5}}+{\mathrm {i} }\sin {\frac {2\pi }{5}},\,\cos {\frac {4\pi }{5}}+{\mathrm {i} }\sin {\frac {4\pi }{5}},\,\cos {\frac {6\pi }{5}}+{\mathrm {i} }\sin {\frac {6\pi }{5}},\,\cos {\frac {8\pi }{5}}+{\mathrm {i} }\sin {\frac {8\pi }{5}}\right)\end{aligned}}}

bzw.

{\displaystyle {}{\begin{aligned}\left(\psi (X),\,\psi (X^{2}),\,\psi (X^{3}),\,\psi (X^{4})\right)&=\left(\zeta ^{2},\,\zeta ^{4},\,\zeta ,\,\zeta ^{3}\right)\\&=\left(\cos {\frac {4\pi }{5}}+{\mathrm {i} }\sin {\frac {4\pi }{5}},\,\cos {\frac {8\pi }{5}}+{\mathrm {i} }\sin {\frac {8\pi }{5}},\,\cos {\frac {2\pi }{5}}+{\mathrm {i} }\sin {\frac {2\pi }{5}},\,\cos {\frac {6\pi }{5}}+{\mathrm {i} }\sin {\frac {6\pi }{5}}\right)\end{aligned}}}

bzw. abgebildet. Die reelle Ganzheitsmatrix ist demnach

${\displaystyle {\begin{pmatrix}\cos {\frac {2\pi }{5}}&\cos {\frac {4\pi }{5}}&\cos {\frac {6\pi }{5}}&\cos {\frac {8\pi }{5}}\\\sin {\frac {2\pi }{5}}&\sin {\frac {4\pi }{5}}&\sin {\frac {6\pi }{5}}&\sin {\frac {8\pi }{5}}\\\cos {\frac {4\pi }{5}}&\cos {\frac {8\pi }{5}}&\cos {\frac {2\pi }{5}}&\cos {\frac {6\pi }{5}}\\\sin {\frac {4\pi }{5}}&\sin {\frac {8\pi }{5}}&\sin {\frac {2\pi }{5}}&\sin {\frac {6\pi }{5}}\end{pmatrix}}.}$