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Let $p$ be a prime number. We say that a number field $F$ satisfies the condition $(H_{p^{n}}^{'})$ when any abelian extension $N / F$ of exponent dividing $p^{n}$ has a normal integral basis with respect to the ring of $p$ -integers. We also say that $F$ satisfies $(H_{p^{\infty}}^{'})$ when it satisfies $(H_{p^{n}}^{'})$ for all $n \geq 1$ . It is known that the rationals $ℚ$ satisfy $(H_{p^{\infty}}^{'})$ for all prime numbers $p$ . In this paper, we give a simple condition for a number field $F$ to satisfy $(H_{p^{n}}^{'})$ in terms of the ideal class group of $K = F (ζ_{p^{n}})$ and a “Stickelberger ideal” associated to the Galois group...

Ideal Class Groups in Basic ...p1 x....x...ps-Extensions of Abelian Number Fields.

Eduardo C. Friedmann (1981/1982)

Inventiones mathematicae

Ideal class groups of cyclotomic number fields I

Franz Lemmermeyer (1995)

Acta Arithmetica

Ideal class groups of cyclotomic number fields II

Franz Lemmermeyer (1998)

Acta Arithmetica

Ideal class groups of cyclotomic number fields III

Franz Lemmermeyer (2008)

Acta Arithmetica

Index of irregularity of a prime.

Ladislav Skula (1980)

Journal für die reine und angewandte Mathematik

Indices of subfields of cyclotomic ℤₚ-extensions and higher degree Fermat quotients

Yoko Inoue, Kaori Ota (2015)

Acta Arithmetica

We consider the indices of subfields of cyclotomic ℤₚ-extensions of number fields. For the nth layer Kₙ of the cyclotomic ℤₚ-extension of ℚ, we find that the prime factors of the index of Kₙ/ℚ are those primes less than the extension degree pⁿ which split completely in Kₙ. Namely, the prime factor q satisfies $q^{p - 1} \equiv 1 (m o d p^{n + 1})$ , and this leads us to consider higher degree Fermat quotients. Indices of subfields of cyclotomic ℤₚ-extensions of a number field which is cyclic over ℚ with extension degree a prime different...

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Displaying 141 – 160 of 461

Generalized Dedekind sums, correlation of L-series, and the Galois module of cot(π/N)cot(mπ/N)

Generalized Iwasawa invariants in a family

Genus number and l-Rank of Genus Group of cyclic extensions of Degree l.

Global units and ideal class groups.

Greenberg's conjecture and cyclotomic towers

Group laws and rings with normal bases.

Groupe de Galois de la p-extension abélienne p-ramifiée maximale d'un corps de nombres.

Groupe des classes de l'algèbre d'un groupe métacyclique.

Groupes de torsion associés à des anneaux d'entiers

Groupes d'unités elliptiques

Hilbert-Speiser number fields and Stickelberger ideals

Ideal Class Groups in Basic ...p1 x....x...ps-Extensions of Abelian Number Fields.

Ideal class groups of cyclotomic number fields I

Ideal class groups of cyclotomic number fields II

Ideal class groups of cyclotomic number fields III

Index of irregularity of a prime.

Indices of subfields of cyclotomic ℤₚ-extensions and higher degree Fermat quotients

Integral differentials and Fermat congruences.

Invariants of the ideal class group and the Hasse norm theorem.

Irregular imaginary fields

Currently displaying 141 – 160 of 461