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Class number one criteria for real quadratic fields. I. (English) Zbl 0625.12002
In this paper, the author intends to generalize results by the reviewer [Class numbers and fundamental units of algebraic number fields, Proc. Int. Conf., Katata/Jap. 1986, 125-137 (1986; Zbl 0612.12010)], S. Louboutin [Arithmétique des corps quadratiques réels et fractions continues, Thèse de Doctorat, Univ. Paris VII] and himself [Proc. Am. Math. Soc. 101, 439-444 (1987)] on class number one problem for real quadratic fields.
Namely, he considers real quadratic fields $${\mathbb{Q}}(\sqrt{n})$$ with positive square-free integers n such that $$n\equiv 1$$ (mod 4) and $$\sqrt{n-1}/2\leq A$$, where A is $$(2T/\sigma -\sigma -1)/U^ 2$$ for the fundamental unit $$(T+U\sqrt{n})/\sigma >1$$ of $${\mathbb{Q}}(\sqrt{n})$$, and gives some necessary and sufficient conditions for these real quadratic fields to have class number one.
Reviewer: H.Yokoi

##### MSC:
 11R11 Quadratic extensions 11R23 Iwasawa theory
##### Keywords:
class number one problem; real quadratic fields
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