By Jürgen Neukirch

Algebraische Zahlentheorie: eine der traditionsreichsten und aktuellsten Grunddisziplinen der Mathematik. Das vorliegende Buch schildert ausführlich Grundlagen und Höhepunkte. Konkret, smooth und in vielen Teilen neu. Neu: Theorie der Ordnungen. Plus: die geometrische Neubegründung der Theorie der algebraischen Zahlkörper durch die "Riemann-Roch-Theorie" vom "Arakelovschen Standpunkt", die bis hin zum "Grothendieck-Riemann-Roch-Theorem" führt.

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Let us call a function in x eventually positive if it takes positive values for all suﬃciently large x. Note that by deﬁnition, if we write f = Ω(g), f = Θ(g), or f ∼ g, it must be the case that f (in addition to g) is eventually positive; however, if we write f = O(g) or f = o(g), then f need not be eventually positive. When one writes “f = O(g),” one should interpret “· = O(·)” as a binary relation between f with g. ” One may also write “O(g)” in an expression to denote an anonymous function f such that f = O(g).

Then we obtain: pe11 µ(d)/d = (1 − 1/p1 ) · · · (1 − 1/pr ). 8) 30 Congruences As another example, suppose f = J. Then we obtain r (1 − 1), µ(d) = (µ J)(n) = i=1 d|n which is 1 if n = 1, and is zero if n > 1. Thus, we have µ J = I. 18 (M¨ obius inversion formula). Let f and F be arithmetic functions. Then we have F = J f if and only if f = µ F . Proof. If F = J f , then µ F = µ (J f ) = (µ J) f = I f = f, and conversely, if f = µ F , then J f =J (µ F ) = (J µ) F = I F = F. ✷ The M¨ obius inversion formula says this: F (n) = f (d) for all positive integers n d|n if and only if µ(d)F (n/d) for all positive integers n.

10, −4, 2, 8, 14, . } [3] = {. . , −9, −3, 3, 9, 15, . } [4] = {. . , −8, −2, 4, 10, 16, . } [5] = {. . , −7, −1, 5, 11, 17, . } 22 Congruences Let us write down the addition and multiplication tables for Z6 . 10. Let n be a positive integer, and consider the set Zn of residue classes modulo n with addition and multiplication of residue classes as deﬁned above. For all α, β, γ ∈ Zn , we have (i) α + β = β + α (addition is commutative), (ii) (α + β) + γ = α + (β + γ) (addition is associative), (iii) α + [0]n = α (existence of additive identity), (iv) α − α = [0]n (existence of additive inverses), (v) α · β = β · α (multiplication is commutative), (vi) (α · β) · γ = α · (β · γ) (multiplication is associative), (vii) α · (β + γ) = α · β + α · γ (multiplication distributes over addition) (viii) α · [1]n = α (existence of multiplicative identity).