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Strukturen und Algebra » Körper und Galois-Theorie » Separabel über Körper mit Charakteristik p
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Universität/Hochschule J Separabel über Körper mit Charakteristik p
LukasNiessen
Aktiv Letzter Besuch: in der letzten Woche
Dabei seit: 30.09.2019
Mitteilungen: 108
Aus: Nordrhein-Westfalen, Bonn, Poppelsdorf
Zum letzten BeitragZum nächsten BeitragZum vorigen BeitragZum erstem Beitrag  Themenstart: 2020-07-04

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Hey,

Sei L/K eine Erweiterung der Char. p > 0.
Man zeige, ein über K algebraisches Element a in L ist genau dann separabel, wenn $K(a)=K(a^p)$ gilt.

----

Ich habe einige Ansätze probiert, insb. mit dem Gradsatz. Aber alles erfolglos.

Hätte jemand einen Hinweis für mich?

Danke!


-----------------
Beste Grüße, Lukas Nießen
PS: Schreibt mir gerne 😄
\(\endgroup\)


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JonyGo
Aktiv Letzter Besuch: in der letzten Woche
Dabei seit: 31.05.2020
Mitteilungen: 27
Zum letzten BeitragZum nächsten BeitragZum vorigen BeitragZum erstem Beitrag  Beitrag No.1, eingetragen 2020-07-04


Für die Körpererweiterung $K(a^p)\subset K(a)=K(a^p)(a)$ ist das Minimalpolynom von $a$ einer Teiler von $X^p-a^p\in K(a^p)[X]$, welches rein inseparabel ist. Daraus folgt direkt $[K(a):K(a^p)]=1\iff K(a)/K(a^p)$ separabel.

Anhand des Körperturm $K(a)/K(a^p)/K$ folgt: $K(a)/K$ separabel $\Rightarrow [K(a):K(a^p)]=1$.

Falls $K(a)/K$ nicht separabel ist, läßt sich das Minimalpolynom von $a$ über $K$ als $\mu=f(X^p)$ schreiben (vgl. Bosch 3.6. Satz 2). Insbesondere gilt $K(a^p)\neq K(a)$.



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LukasNiessen
Aktiv Letzter Besuch: in der letzten Woche
Dabei seit: 30.09.2019
Mitteilungen: 108
Aus: Nordrhein-Westfalen, Bonn, Poppelsdorf
Zum letzten BeitragZum nächsten BeitragZum vorigen BeitragZum erstem Beitrag  Beitrag No.2, vom Themenstarter, eingetragen 2020-07-15

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Danke für die Antwort!

Der erste Teil ist doch, nur anders formuliert:

Für die Körpererweiterung $K(a^p)\subset K(a)=K(a^p)(a)$ ist das Minimalpolynom von $a$ ein Teiler von $X^p-a^p\in K(a^p)[X]$, welches rein inseparabel ist, damit ist also a inseparabel über $K(a^p)$. Wenn a nun separabel über K ist, ist es dies auch über $K(a^p)$ und es folgt $a \in K(a^p)$ und damit $K(a)\subset K(a^p)$ und damit die Gleichheit.

Oder?




-----------------
Beste Grüße, Lukas Nießen
PS: Schreibt mir gerne 😄
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