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On a comparison of the 4-ranks of the narrow ideal class groups of \(\mathbb{Q}(\sqrt m)\) and \(\mathbb{Q}(\sqrt{-m})\). (English) Zbl 0921.11055
For a square-free integer \(m\), let \(r^+_4(m)\) be the \(4\)-rank of the narrow ideal class group of \(\mathbb Q(\sqrt{m})\). It is well known that if \(m > 0\), then \[ r^+_4(m) \leq r^+_4(-m) \leq r^+_4(m) +1, \] and there exist several proofs of this result. The author provides a further proof, based on an old idea of L. Rédei and H. Reichardt [J. Reine Angew. Math. 170, 69-74 (1933; Zbl 0007.39602)] from 1933. He also gives criteria for equality in several special cases.
MSC:
11R29 Class numbers, class groups, discriminants
11R11 Quadratic extensions
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