Algebraische Zahlentheorie [Lecture notes] by Jakob Stix PDF

By Jakob Stix

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12. Sei A ein Dedekindring und p ein maximales Primideal. Wir definieren die Abbildung vp : IA → Z I → vp (I) = min{vp (x) ; x ∈ I}. 13. 9 ist. Insbesondere ist vp ein Gruppenhomomorphismus und Ip = (π vp (I) ). Hier ist π ∈ Ap eine Uniformisierende der diskreten Bewertung vp . 14. Zu gebrochenen Idealen I, J ⊆ K definieren wir das gebrochene Ideal (I : J) = {x ∈ K ; xJ ⊆ I} = ker(K → HomA (J, K/I)). 15. Seien A ein Dedekindring und p ein maximales Primideal. Die Bewertung von gebrochenen Idealen hat die folgenden Eigenschaften: für alle gebrochenen Ideale I, J und alle x ∈ K × gilt: Algebraische Zahlentheorie (i) (ii) (iii) (iv) (v) 35 vp (IJ) = vp (I) + vp (J), vp (I + J) = min{vp (I), vp (J)}, vp (I ∩ J) = max{vp (I), vp (J)}, vp ((I : J)) = vp (I) − vp (J), vp ((x)) = vp (x).

Vr R = Γ R und Γ 0 = Γ ∩ W0 . Dann ist Γ0 ein Gitter in W0 und zwangsläufig dim Γ0 R ≤ dim W0 = r − 1. Der Satz gilt somot für Γ0 . Seien γ1 , . . , γr−1 eine Z-Basis von Γ0 aus R-linear unabhängigen Vektoren. Dann kann man die v1 , . . , vr−1 durch die γ1 , . . , γr−1 ersetzen. Wir betrachten nun pr : Γ ⊆ W → W/W0 = R · vr . ¯ Das Bild Γ = pr(Γ) ⊆ R ist ein Gitter, weil für alle 0 < c ∈ R ¯ ∩ {avr ; |a| < c} = pr (Γ ∩ (W0 × (−c, c) · vr ) Γ r−1 ai γi ; |ai | ≤ 1, |a| ≤ c} = pr Γ ∩ {x = avr + i=1 Weil Γ diskret und {x = avr + r−1 i=1 ai γi ; |ai | ≤ 1, |a| ≤ c} beschränkt ist, handelt es sich um ¯ diskret in R · vr R.

Also nimmt x, y Werte in R an. Weiter gilt x, x > 0 für x = 0, weil −, − hermitesches Skalarprodukt ist. 48 JAKOB STIX Die Abbildung jC induziert einen Isomorphismus auf Gal(C/R)-Invarianten: + ∼ jR : F ⊗Q R − → . 12. Sei F ein Zahlkörper. Dann bildet + ∼ j : F → F ⊗Q R − → C τ jede Ordnung und jedes gebrochene Ideal von F auf ein vollständiges Gitter des Minkowski– Raumes ab. Beweis. Für ein gebrochenes Ideal I ⊆ F mit s ∈ F × so daß sI = a ⊆ oF , folgt I ⊗Z Q = a ⊗Z Q = oF ⊗Z Q = F, denn a hat endlichen Index in oF .

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Algebraische Zahlentheorie [Lecture notes] by Jakob Stix


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