Algebra I by Walter Gubler

By Walter Gubler

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Extra resources for Algebra I

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Dann gilt: ∼ Z/nZ −→ (Z/q1 Z) × . . × (Z/qr Z) k (mod n) → (k (mod q1 ), . . , k (mod qr )) Beweis. 4 (Klassifikation abelscher Gruppen). Jede endliche abelsche Gruppe G ist isomorph zu genau einer Gruppe der Form (Z/q1 Z) × . . 1) wobei q1 , . . , qs nicht notwendigerweise verschiedene Primzahlpotenzen sind. 1) der Isomorphietyp der Gruppe G. Beweis. Sei G eine endliche abelsche Gruppe. 2 gilt: G∼ = (Z/n1 Z) × . . 2) Sei ni = pi1 vi1 · . . · piri viri die Primfaktorzerlegung von ni . Nach dem chinesischen Restsatz gilt: Z/ni Z ∼ = (Z/pi1 vi1 Z) × .

Weiter sei p ein irreduzibles Element aus R mit p an , p | ai ∀i < n und p2 a0 . Dann ist f (x) irreduzibel in Q[x]. Beweis. Wir betrachten wieder den surjektiven Ringhomomorphismus Φp : R[x] −→ (R/pR)[x], αk xk → αk xk . 7 ist pR = p ein Primideal in R. 12, dass R/ p ein Integrit¨atsbereich ist. Also ist R/ p ein Teilring seines Quotientenk¨ orpers, den wir hier mit F bezeichnen. 7). 12). Das stimmt aber nicht f¨ ur jeden faktoriellen Ring! Wir erinnern daran, dass der Inhalt µ(f ) von f gleich dem ggT der Koeffizienten a0 , .

Wir bezeichnen die Inverse von a bzgl ” + ” mit −a und setzen a − b := a + (−b). Wir benutzen folgende Rechenregeln: (1) a · 0 = 0 = 0 · a (2) Das Einselement ist eindeutig. (3) −a = (−1) · a (d) a · (b − c) = a · b − a · c und (b − c) · a = b · a − c · a 33 34 KAPITEL 2. RINGTHEORIE Beweis. Folgt alles sofort aus den Axiomen. 4. Ein K¨ orper ist ein kommutativer Ring so, dass K\ {0} eine Gruppe bzgl ” · ” ist. 5. • Z ist ein kommutativer Ring • Die n × n Matrizen mit Eintr¨agen aus dem K¨orper K bilden einen Ring der nicht kommutativ ist (f¨ ur n ≥ 2).

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