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On ideal class groups of cyclotomic fields

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On the class number of p -th cyclotomic field

Archiv Der Mathematik, 2000
Let \(p\) denote an odd prime, and let \(h^-(p)\) denote the minus class number of the field of \(p\)-th roots of unity. It is known that \[ \log h^-(p) = \frac{p-3}4 \log p - \frac{p}2 \log 2\pi + \log(1-\beta) + f(1), \] where \(\beta\) denotes a Siegel zero if it exists and if \(p \equiv 3 \bmod 4\) (and the term involving \(\beta\) does not appear ...
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A Class Number Formula for Cyclotomic Fields

The Annals of Mathematics, 1962
where En+ is the group of units in F,+ and To is the subgroup of circular units in E.+.' Let Gn denote the Galois group of Fn over Q and let 9R = Z[Gn] be the group ring of Gn over the ring of rational integers Z. In the present paper, we shall first transform the formula (1) and show that the first factor halso can be expressed as a group index of ...
Ichiro Satake, Genjiro Fujisaki, Kazuya Kato, Masato Kurihara, Shoichi Nakajima   +4 more
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The Construction of Regular Hadamard Matrices by Cyclotomic Classes

Bulletin of the Iranian Mathematical Society, 2020
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Tianbing Xia, Mingyuan Xia, Jennifer Seberry   +2 more
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Unit Fractions and the Class Number of a Cyclotomic Field

Journal of the London Mathematical Society, 2002
For a prime number \(p\), let \(h(p)\) denote the class number of the \(p\)th cyclotomic field \({\mathbb Q}(e^{2\pi i/p})\) and \(h_2(p)\) denote the class number of its maximal real subfield \({\mathbb Q}(\cos(2\pi /p))\). Kummer showed that the ratio \(h_1(p)=h(p)/h_2(p)\) is an integer which he called the first factor of the class number.
Croot, Ernest S. III, Granville, Andrew
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Class Number of Cyclotomic Fields

2001
In this chapter we shall derive formulas for the class number of cyclotomic fields generated by pth roots of unity, where p is a prime. They involve L-series and Gaussian sums associated to characters.
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Class numbers of real cyclotomic fields

Publicationes Mathematicae Debrecen, 2004
Using simplest sextic fields, the author shows that there are real cyclotomic fields of class numbers greater than their conductors. More precisely, he proves the following theorem: Assume that \(\Delta_m=m^2+3m+9\equiv1\pmod4\) is square-free, \(m\geq-1\). Let \(t_m\) denote the number of distinct prime factors of \(\Delta_m\).
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Cyclotomic Fields of Class Number One

1982
In this chapter we determine those m for which \( \mathbb{Q}({\zeta _m}) \) has class number one. In Chapter 4, the Brauer–Siegel theorem was used to show that there are only finitely many such fields, but the result was noneffective: there was no computable bound on m. So we need other techniques.
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Asymptotic Results for Class Number Divisibility in Cyclotomic Fields

Canadian Mathematical Bulletin, 1983
AbstractLet n ≥3 and m≥3 be integers. Let Kn be the cyclotomic field obtained by adjoining a primitive nth root of unity to the field of rational numbers. Let denote the maximal real subfield of Kn. Let hn (resp., ) denote the class number of Kn (resp., ). For fixed m we show that m divides hn and hn for asymptotically almost all n.
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Class Group Relations in Cyclotomic Fields

American Journal of Mathematics, 1952
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