Algebra II [Lecture notes] by Christoph Schweigert

By Christoph Schweigert

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Extra resources for Algebra II [Lecture notes]

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Setze τ := ρ1 ρ2 ∈ G(E1 E2 /K). Man rechnet nach: τ|E1 = (ρ1 )|E1 (ρ2 )|E1 = σ1 idE1 = σ1 , und entsprechend auch τ|E2 = σ2 . Also ist der angegebene Gruppenhomomorphismus auch surjektiv. 4. (i) Sei K ein K¨orper und M ein Monoid. Ein Charakter von M mit Werten in K ist ein Homomorphismus von Monoiden χ : M → K× . Ein Charakter heißt trivial, wenn χ(m) = 1 f¨ ur alle m ∈ M gilt. 46 (ii) Die Funktionen f : M → K bilden einen K-Vektorraum. Entsprechend heißen Funktionen fi : M → K, i = 1, . . , n linear unabh¨angig u ¨ber K, falls aus a1 f1 + · · · + an fn = 0 mit ai ∈ K folgt, dass a1 = a2 = · · · = an = 0 gilt.

X (p−1)p , Fpm (X) = p X −1 m−1 m−1 denn Fpm (X) teilt 1 + X p + · · · + X (p−1)p : jede Einheitswurzel der Ordnung pm ist Nullstelle des Z¨ahlers, aber nicht des Nenners. Außerdem stimmen die Grade der Polynome u ¨berein: grad Fpm = ϕ(pm ) = pm−1 (p − 1) . 1 Kreisteilungsko ats¨rper, quadratisches Reziprozit¨ gesetz n-Teilung des Kreises Wir wollen noch einmal auf das Problem der Konstruierbarkeit mit Zirkel und Lineal zur¨ uckkommen. 1. Sei M ⊆ C eine Untermenge, die 0, 1 enth¨alt und K der unter komplexer Konjugation abgeschlossene K¨orper K = Q(M ∪ M ).

Betrachte S := χ(σ)σ(ζ) = ν∈(Z/p)× ν∈G(Q(ζ)/Q) ν ν ζ ∈ Q(ζp ) . p Dann gilt S2 = p−1 −1 p = (−1) 2 p =: p∗ p f¨ ur p = 2 . √ ur p = 2 ist Q( p∗ ) der eindeutig bestimmte quadratische Erweite(ii) F¨ rungsk¨orper von Q in Q(ζp ). 40 Beweis. (ii) folgt unmittelbar aus (i): da S ∈ Q(ζp ) liegt und S 2 = p∗ gilt, ist √ √ √ ± p∗ ∈ Q(ζp ). Andererseits liegt p∗ ∈ Q, also ist Q( p∗ ) der gesuchte K¨orper. (i) folgt aus der folgenden Rechnung: µ ν+µ ζ = p ν p S2 = ν,µ ν,µ νµ ν+µ ζ p Ersetze nun die Summationsvariable ν durch µν.

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