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# Kreisteilungskörper/Q/9/Permutationen/Automorphismen/Aufgabe

Es sei ${\displaystyle {}\zeta =e^{2\pi {\mathrm {i} }/9}}$. Bestimme, ob die folgenden Permutationen auf der Menge der primitiven neunten Einheitswurzeln ${\displaystyle {}\zeta ,\zeta ^{2},\zeta ^{4},\zeta ^{5},\zeta ^{7},\zeta ^{8}}$ durch einen Körperautomorphismus des neunten Kreisteilungskörpers herrühren.

1.  ${\displaystyle {}x}$ ${\displaystyle {}\zeta }$ ${\displaystyle {}\zeta ^{2}}$ ${\displaystyle {}\zeta ^{4}}$ ${\displaystyle {}\zeta ^{5}}$ ${\displaystyle {}\zeta ^{7}}$ ${\displaystyle {}\zeta ^{8}}$ ${\displaystyle {}\sigma (x)}$ ${\displaystyle {}\zeta ^{2}}$ ${\displaystyle {}\zeta ^{4}}$ ${\displaystyle {}\zeta ^{8}}$ ${\displaystyle {}\zeta }$ ${\displaystyle {}\zeta ^{7}}$ ${\displaystyle {}\zeta ^{5}}$
2.  ${\displaystyle {}x}$ ${\displaystyle {}\zeta }$ ${\displaystyle {}\zeta ^{2}}$ ${\displaystyle {}\zeta ^{4}}$ ${\displaystyle {}\zeta ^{5}}$ ${\displaystyle {}\zeta ^{7}}$ ${\displaystyle {}\zeta ^{8}}$ ${\displaystyle {}\sigma (x)}$ ${\displaystyle {}\zeta ^{8}}$ ${\displaystyle {}\zeta ^{7}}$ ${\displaystyle {}\zeta ^{5}}$ ${\displaystyle {}\zeta ^{4}}$ ${\displaystyle {}\zeta ^{2}}$ ${\displaystyle {}\zeta }$
3.  ${\displaystyle {}x}$ ${\displaystyle {}\zeta }$ ${\displaystyle {}\zeta ^{2}}$ ${\displaystyle {}\zeta ^{4}}$ ${\displaystyle {}\zeta ^{5}}$ ${\displaystyle {}\zeta ^{7}}$ ${\displaystyle {}\zeta ^{8}}$ ${\displaystyle {}\sigma (x)}$ ${\displaystyle {}\zeta }$ ${\displaystyle {}\zeta ^{2}}$ ${\displaystyle {}\zeta ^{4}}$ ${\displaystyle {}\zeta ^{5}}$ ${\displaystyle {}\zeta ^{7}}$ ${\displaystyle {}\zeta ^{8}}$
4.  ${\displaystyle {}x}$ ${\displaystyle {}\zeta }$ ${\displaystyle {}\zeta ^{2}}$ ${\displaystyle {}\zeta ^{4}}$ ${\displaystyle {}\zeta ^{5}}$ ${\displaystyle {}\zeta ^{7}}$ ${\displaystyle {}\zeta ^{8}}$ ${\displaystyle {}\sigma (x)}$ ${\displaystyle {}\zeta ^{5}}$ ${\displaystyle {}\zeta }$ ${\displaystyle {}\zeta ^{2}}$ ${\displaystyle {}\zeta ^{4}}$ ${\displaystyle {}\zeta ^{8}}$ ${\displaystyle {}\zeta ^{7}}$