Class numbers of quadratic forms over real quadratic fields
نویسندگان
چکیده
منابع مشابه
Perfect unary forms over real quadratic fields
Let F = Q( √ d) be a real quadratic field with ring of integers O. In this paper we analyze the number hd of GL1(O)orbits of homothety classes of perfect unary forms over F as a function of d. We compute hd exactly for square-free d ≤ 200000. By relating perfect forms to continued fractions, we give bounds on hd and address some questions raised by Watanabe, Yano, and Hayashi.
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Introduction. Witt [5] proved that two binary or ternary quadratic forms, over an arbitrary field (of characteristic not 2) are equivalent if and only if they have the same determinant and Hasse invariant. His proof is brief and elegant but uses a lot of the theory of simple algebras. The purpose of this note is to make this fundamental theorem more accessible by giving a short proof using only...
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1. The Hasse Principle(s) For Quadratic Forms Over Global Fields 1 1.1. Reminders on global fields 1 1.2. Statement of the Hasse Principles 2 2. The Hasse Principle Over Q 3 2.1. Preliminary Results: Reciprocity and Approximation 3 2.2. n ≤ 1 6 2.3. n = 2 6 2.4. n = 3 6 2.5. n = 4 8 2.6. n ≥ 5 9 3. The Hasse Principle Over a Global Field 9 3.1. n = 2 10 3.2. n = 3 10 3.3. n = 4 11 3.4. n ≥ 5 12...
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In this paper an unconditional probabilistic algorithm to compute the class number of a real quadratic field Q( √ d) is presented, which computes the class number in expected time O(d1/5+ ). The algorithm is a random version of Shanks’ algorithm. One of the main steps in algorithms to compute the class number is the approximation of L(1, χ). Previous algorithms with the above running time O(d1/...
متن کاملIndivisibility of Class Numbers of Real Quadratic Fields
Although the literature on class numbers of quadratic fields is quite extensive, very little is known. In this paper we consider class numbers of real quadratic fields, and as an immediate consequence we obtain an estimate for the number of vanishing Iwasawa λ invariants. Throughout D will denote the fundamental discriminant of the quadratic number fieldQ( √ D), h(D) its class number, and χD :=...
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ژورنال
عنوان ژورنال: Nagoya Mathematical Journal
سال: 1977
ISSN: 0027-7630,2152-6842
DOI: 10.1017/s0027763000017736