By Heinz-Georg Quebbemann

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Sn ) heiße f . Es hat L als Zerf¨allungsk¨orper, daher ist der Grad von L u ¨ber dem Teilk¨orper K0 (s1 , . . , w¨ahrend aber schon [L : E] = ord Sn = n! gilt. 3 Der K¨orper E der symmetrischen Funktionen ist K0 (s1 , . . , sn ). Nach dem Polynom f mit ”variablen Nullstellen” betrachten wir nun eines mit ”variablen Koeffizienten”. 4 Sei K = K0 (u1 , . . , un ) der rationale Funktionenk¨orper in n unabh¨angigen Variablen u1 , . . , un . Das allgemeine Polynom n-ten Grades g(t) := tn + u1 tn−1 + .

Sn ). Nach dem Polynom f mit ”variablen Nullstellen” betrachten wir nun eines mit ”variablen Koeffizienten”. 4 Sei K = K0 (u1 , . . , un ) der rationale Funktionenk¨orper in n unabh¨angigen Variablen u1 , . . , un . Das allgemeine Polynom n-ten Grades g(t) := tn + u1 tn−1 + . . + un ∈ K[t] ist separabel und hat u ¨ber K die Galoisgruppe Sn . Zum Beweis gen¨ ugt es zu zeigen, dass mit den elementar-symmetrischen s1 , . . , sn ∈ K0 [t1 , . . , tn ] der Einsetzungshomomorphismus ψ : K0 [u1 , .

Diese Bestimmung wird jetzt aber f¨ ur den K¨orper E durchgef¨ uhrt. Spezielle Elemente von E sind die elementar-symmetrischen Polynome n s1 = n ti , i=1 s2 = ti tj , . . , sn = 1≤i

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