J. Sander et al.'s Algebraische Zahlentheorie [Lecture notes] PDF

By J. Sander et al.

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H. OF ist norm-euklidisch). Beweis: Wir setzen   2 f¨ ur ε :=  1 f¨ ur D ≡ 1 mod 4, D ≡ 2, 3 mod 4. Offenbar l¨asst sich jedes σ ∈ F schreiben als σ = r1 + r2 √ D ε r1 , r 2 ∈ Q . 50 f¨ ur norm-euklidische Ringe ist ¨aquivalent zu: F¨ ur √ alle σ ∈ Q( D) existiert ein β ∈ OF mit |NF (σ − β)| < 1 . 43 haben wir also ein √ 1 β = (x + y D) ∈ OF ε (x, y ∈ Z) zu finden derart, dass (∗) |NF (σ − β)| = r1 − x ε 2 − 1 (r2 − y)2 D < 1 . ε2 Wir nehmen an, dass (∗) bei gegebenem r1 , r2 ∈ Q f¨ ur alle x, y ∈ Z verletzt ist.

Eine primitive p-te Einheitswurzel, also p−1 NQ(ξ) (1 − ξ) = j=1 (1 − ξ j ) = Φp (1) = p wegen Φp (x) = (xp − 1)/(x − 1) = xp−1 + xp−2 + · · · + x + 1. 22 (ii) kommt wegen NQ(ζ) (−1) = ±1 NQ(ζ) (ξ − 1) = ±(NQ(ξ) (ξ − 1))p a−1 a−1 = ±pp . Da auch NQ(ζ) (ζ −1 ) = ±1, erhalten wir aus (∗) a−1 NQ(ζ) (Φpa (ζ)) · (±pp a a −pa−1 ) ) = NQ(ζ) (pa ) = (pa )ϕ(p ) = pa(p . 6 discr(B1 ) = ±NQ(ζ) (mζ,Q (ζ)) = ±NQ(ζ) (Φpa (ζ)) = ±pp a−1 (ap − a − 1) . Dies beweist die Zwischenbehauptung, da n ≥ 3. a )−1 Wir setzen η := 1 − ζ.

Ii) =⇒“ ” Seien alle Zerlegungen eindeutig. F¨ ur α ∈ OF irreduzibel ist zu zeigen: α ist prim. h. es gibt σ ∈ OF mit βγ = ασ. Nach Voraussetzung haben β, γσ eindeutige Zerlegungen r β =u· t s βj j=1 , γ=v· γj , j=1 σ=w· σj j=1 mit u, v, w ∈ UF und βj , γj , σj alle irreduzibel. Also t α·w· r σj = ασ = βγ = uv j=1 j=1 s βj · γj . j=1 Da α irreduzibel ist, folgt aus der eindeutigen Faktorisierung, dass α ∈ {βj : 1 ≤ j ≤ r} ∩ {γj : 1 ≤ j ≤ s}. h. α ist prim. ⇐=“ ” Sei jedes irreduzible Element von OF prim.

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Algebraische Zahlentheorie [Lecture notes] by J. Sander et al.


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