We give an algebraic proof of a class number formula for dihedral extensions of number fields of degree 2q, where q is any odd integer. Our formula expresses the ratio of class numbers as a ratio of orders of cohomology groups of units and allows one to recover similar formulas which have appeared in the literature. As a corollary of our main result, we obtain explicit bounds on the (finitely many) possible values which can occur as ratio of class numbers in dihedral extensions. Such bounds are obtained by arithmetic means, without resorting to deep integral representation theory.

中文翻译：

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二面角扩展的类号公式

我们给出了 2 次数域的二面扩展的类数公式的代数证明q， 在哪里q是任何奇数。我们的公式将类别数的比率表示为单位的上同调群的阶数之比，并允许人们恢复文献中出现的类似公式。作为我们主要结果的推论，我们获得了（无限多）可能值的明确界限，这些可能值可以作为二面扩展中类数的比率出现。这样的界限是通过算术手段获得的，而不是求助于深度积分表示理论。