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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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