Algebraic Geometry

Advanced Euclidean Geometry (Dover Books on Mathematics) - download pdf or read online

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By Roger A. Johnson

This vintage textual content explores the geometry of the triangle and the circle, targeting extensions of Euclidean concept, and studying intimately many rather contemporary theorems. a number of hundred theorems and corollaries are formulated and proved thoroughly; a number of others stay unproved, for use through scholars as workouts. 1929 variation.

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Extra info for Advanced Euclidean Geometry (Dover Books on Mathematics)

Example text

Dann gilt: (i) Jedes α ∈ OF , α = 0, l¨asst sich in ein Produkt irreduzibler Faktoren zerlegen. (ii) Jedes α ∈ OF , α = 0, besitzt eine bis auf Reihenfolge der Faktoren und Assoziierte eindeutige solche Zerlegung genau dann, wennn jedes irreduzible Element von OF prim ist. 1 ALGEBRAISCHE ZAHLEN 44 Beweis: ur gewisse β, γ ∈ OF \ UF . (i) Ist α nicht selbst irreduzibel, so gilt α = β · γ f¨ Iteration dieses Zerlegungsprozesses liefert die gew¨ unschte Faktorisierung. Der Prozess ist endlich, denn: Ist N (δ) = ±1 f¨ ur ein δ ∈ OF , so gilt 1 = ±N (δ) = δ · ((±1) · δ2 · δ3 · δ4 · .

Da βs nach Voraussetzung prim (also keine Einheit) ist, impliziert βs | uα1 · . . A. βs | αr . Damit sind βs und αr assoziiert. Da OF Integrit¨atsring ist (OF besitzt keine Nullteiler, denn C besitzt keine Nullteiler), k¨onnen wir urzen und erhalten βs = wαr , w ∈ UF , in (∗) k¨ u α1 · . . · αr−1 = vβ1 · . . · βs−1 Induktion liefert die Behauptung. 48 Sei D ein Integrit¨atsbereich, in dem jedes Element = 0 eindeutig in irreduzible Elemente zerf¨allt. Dann heißt D ein ZPE-Ring (Zerlegung in Primelemente eindeutig).

35 haben wir (∗) discr (B2 ) = D2 · discr (B1 ) f¨ ur das dort ausgegebene D, in unserer Situation D ∈ Z. 37 auch die beiden Diskriminanten ganzrational sind, folgt discr (B1 ) | discr (B2 ). Durch Vertauschen der Rollen ergibt sich auf die gleiche Weise discr (B2 ) | discr (B1 ). Also discr (B1 ) = ±discr (B2 ) , wobei das Minuszeichen wegen (∗) nicht m¨oglich ist. ✷ 1 ALGEBRAISCHE ZAHLEN 37 Das vorstehende Korollar besagt im Wesentlichen, dass die Diskriminante einer Ganzheitsbasis eines Zahlk¨orpers F eine Invariante von F ist.

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