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  Galois-Gruppe bestimmen
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Hallo!
Es geht nochmal darum eine Galois-Gruppe zu bestimmen, und zwar folgenden Polynoms:
$X^8 - 2 \in \IQ[X]$.
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Der Zerfällungskörper ist $\IQ(\sqrt[8]{2}, i)$ und die Erweiterung hat den Grad 16.
Es gilt:
$\text{Gal}(\IQ(\sqrt[8]{2}, i) / \IQ) \subset S_8$.
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Ich komme hier aber nicht weiter, wie kann ich nun die 16 Automorphismen der Galois-Gruppe bestimmen?
Danke!
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Beste Grüße, Lukas Nießen
PS: Schreibt mir gerne 😄\(\endgroup\)
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Hallo,
ich glaube nicht, das das so richtig
ist. Müsste nicht die prim.8. Einheitswuzel
mit im zerfällunskörper sein?
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Sorry,
das war ein Denkfehler, der zerfällungskörper
war doch richtig...
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Ich habe einen Artikel geschrieben, wie man solche einfachen Galoisgruppen berechnet: article.php?sid=1779
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@ollie3: Das stimmt schon, aber die primitive 8. Einheitswurzel ist $$\sqrt{i} = \frac{1+i}{\sqrt{2}} = \frac{1+i}{\sqrt[8]{2}^4},$$sodass man genauso gut $i$ als Erzeuger nehmen kann.
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Perfekt, danke euch!
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Beste Grüße, Lukas Nießen
PS: Schreibt mir gerne 😄
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LukasNiessen hat die Antworten auf ihre/seine Frage gesehen.
LukasNiessen hat selbst das Ok-Häkchen gesetzt. |  [Neues Thema]  [Druckversion] |
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Themenstart: 2020-10-23
Beitrag No.3, eingetragen 2020-10-23