Algebra I by Walter Gubler

By Walter Gubler

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Additional info for Algebra I

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Am Schluss zeigen wir noch ein Irreduzibilit¨atskriterium f¨ ur Polynome. Die Irreduzibilit¨at von Polynomen ist schwierig zu pr¨ ufen und das Kriterium von Eisenstein ist oft hilfreich. Zur Erinnerung: a, b ∈ R heißen assoziiert ⇐⇒ a = ub f¨ ur ein u ∈ R∗ . ¨ Wir notieren dann a ∼ b und dies ist eine Aquivalenzrelation. Weiter sei P ein Repr¨ asentantensystem der irreduziblen Elemente in R. 7. 1. Sei α ∈ Q\ {0}. Dann gibt es f¨ ur jedes p ∈ P genau ein vp (α) ∈ Z so, dass vp (α) = 0 bis auf endlich viele p ∈ P und so, dass pvp (α) α=u p∈P f¨ ur ein eindeutig bestimmtes u ∈ R∗ .

RINGTHEORIE Wenn wir weiter machen mit dem euklidischen Algorithmus, erhalten wir 1188 = 4 · 297 + 0. Mit x3 = 4x2 + y2 und y3 = x2 erhalten wir 297x3 = 297 + 0y3 = 594 als letzte Gleichung und es gilt f¨ ur den gr¨ossten gemeinsamen Teiler ggT(10098, 1485) = 297. Da 297 ein Teiler von c = 594 ist, k¨onnen wir die diophantische Gleichung l¨ osen. Nach K¨ urzen erhalten wir die allgemeine L¨osung der letzten Gleichung mit y3 ∈ Z beliebig und x3 = 2. R¨ uckw¨artseinsetzen liefert x2 = y3 und y2 = 2 − 4x2 = 2 − 4y3 .

I) I R Hauptideal :⇐⇒ I wird von einem Element (ii) R heißt Hauptidealbereich :⇐⇒ jedes Ideal ist Hauptideal und R ist ein Integrit¨ atsbereich. 4. 4). 5. Seien g, g ∈ R. Dann gilt: (i) g ⊆ g ⇐⇒ g | g (ii) g = g ⇐⇒ g ∈ R∗ g , falls R ein Integrit¨ atsbereich ist. Beweis. • zu (i): g ⊆ g ⇐⇒ g ∈ g ⇐⇒ ∃a ∈ R : g = ag ⇐⇒ g | g (i) • zu (ii): ” =⇒ ”: g = g =⇒ ∃u, v ∈ R : g = ug und g = vg =⇒ g = uvg =⇒ (1 − uv)g = 0. Falls g = 0, dann gilt auch g = 0 und die Behauptung ist trivial. Falls g = 0, dann muss 1 = uv gelten, weil R ein Integrit¨ atsbereich ist.

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