By Walter Gubler

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Wegen d(a1 ) > d(a2 ) > d(a3 ) > ... ≥ 0 terminiert der Algorithmus in endlich vielen Schritten. Das heißt es gibt ein n ∈ N mit an+1 = 0 und damit ggT(a, b) = ggT(a0 , a1 ) = . . = ggT(an−1 , an ) = an . 9. Berechne zuerst ggT(42, 15): 52 KAPITEL 2. RINGTHEORIE • 42 = 2 · 15 + 12 • 15 = 1 · 12 + 3 • 12 = 4 · 3 + 0 =⇒ ggT(12, 3) = 3 =⇒ ggT(42, 15) = 3 Als weiteres Beispiel berechnen wir ggT(25326, 1555): i 1 2 3 4 5 ai−1 25326 1555 446 217 12 ai 1555 446 217 12 1 qi 16 3 2 18 12 ai+1 446 217 12 1 0 Somit gilt ggT(25326, 1555) = 1.

M gen¨ ugt der Teilerkettenbedingung :⇐⇒ Es gibt keine unendliche Kette . . | an | an−1 | . . | a1 von paarweise nicht assoziierten Elementen ai ∈ M . 8. Jedes faktorielle Monoid erf¨ ullt die Teilerkettenbedingung. Beweis. Wir betrachten eine Teilerkette wie oben und setzen a := a1 . Wenn a eine Einheit ist, dann ist jeder Teiler von a eine Einheit und es kann keinen nicht assoziierten Teiler a2 von a1 geben. Falls a keine Einheit ist, dann haben wir die eindeutige Faktorisierung a = p1 · · · ps in irreduzible Elemente.

Wir nehmen K-linear unabh¨angige Elemente β1 , . . , βl aus L und Llinear unabh¨ angige Elemente γ1 , . . , γm aus M . Nach dem ersten Schritt sind dann (βi γj )1≤i≤l,1≤j≤m K-linear unabh¨angig in M , also folgt [M : K] ≥ ml. 6. Folgendes wichtige Verfahren konstruiert zu einem gegeben irreduziblen Polynom f (x) ∈ K[x] eine K¨orpererweiterung L von K, die eine Nullstelle von f (x) enth¨ alt. Wir setzen L := K[x]/ f (x) . 1 ist L ein K¨orper, weil f (x) irreduzibel ist. Wir haben einen K¨orperhomomorphismus K −→ L, α → α := α + f (x) , wobei wir hier α als das konstante Polynom α ansehen.

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Algebra I by Walter Gubler
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