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