On the Class Number of Abelian Number Fields by Helmut Hasse

Author:Helmut Hasse
Language: eng
Format: epub
ISBN: 9783030015121
Publisher: Springer International Publishing

(3.8.6)

By this formula the relative class number h ∗ of K∕K 0 for the considered field K is expressed in the form66

(3.8.7)

In the mentioned special field , instead of the formulas above, it holds that h = Q h 0 h 1 h 2 and h ∗ = Q h 1 h 2. In this case, as determined in Theorem 3.27, Q = 1 and hence h = h 0 h 1 h 2 and h ∗ = h 1 h 2. Since here h 0, h 1, h 2 = 1 are known, it also follows that h = 1 and h ∗ = 1. For simplicity we exclude in the following consideration this exceptional case, which is no longer interesting.

The various forms in which formula (3.8.6) for the product of class numbers appears in the literature are distinguished from our point of view by how exactly and in what way the unit index Q = 1 or 2 of K∕K 0 is determined. In the following we will summarize all the statements on Q that arise by the general theory of the unit index developed in Sects. 3.2–3.7 for the special fields considered here.

The two cases for w to be distinguished by the general theory are determined here in the following way: . w≢0 (mod 4), when f 1 ≠ 4, f 2 ≠ 4.

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