Kreisteilungskörper/Q/9/Permutationen/Automorphismen/Aufgabe

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

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

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

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

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

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