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

Yoko Inoue; Kaori Ota

Displaying similar documents to “Indices of subfields of cyclotomic ℤₚ-extensions and higher degree Fermat quotients”

On the subfields of cyclotomic function fields

Zhengjun Zhao, Xia Wu (2013)

Czechoslovak Mathematical Journal

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Let $K = 𝔽_{q} (T)$ be the rational function field over a finite field of $q$ elements. For any polynomial $f (T) \in 𝔽_{q} [T]$ with positive degree, denote by $Λ_{f}$ the torsion points of the Carlitz module for the polynomial ring $𝔽_{q} [T]$ . In this short paper, we will determine an explicit formula for the analytic class number for the unique subfield $M$ of the cyclotomic function field $K (Λ_{P})$ of degree $k$ over $𝔽_{q} (T)$ , where $P \in 𝔽_{q} [T]$ is an irreducible polynomial of positive degree and $k > 1$ is a positive divisor of $q - 1$ . A formula for the analytic class...

Class group of a cyclotomic $ℤ_{p} \times ℤ_{ℓ}$ -extension

Humio Ichimura (2011)

Acta Arithmetica

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On the ring of $p$ -integers of a cyclic $p$ -extension over a number field

Humio Ichimura (2005)

Journal de Théorie des Nombres de Bordeaux

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Let $p$ be a prime number. A finite Galois extension $N / F$ of a number field $F$ with group $G$ has a normal $p$ -integral basis ( $p$ -NIB for short) when $𝒪_{N}^{'}$ is free of rank one over the group ring $𝒪_{F}^{'} [G]$ . Here, $𝒪_{F}^{'} = 𝒪_{F} [1 / p]$ is the ring of $p$ -integers of $F$ . Let $m = p^{e}$ be a power of $p$ and $N / F$ a cyclic extension of degree $m$ . When $ζ_{m} \in F^{\times}$ , we give a necessary and sufficient condition for $N / F$ to have a $p$ -NIB (Theorem 3). When $ζ_{m} \notin F^{\times}$ and $p ∤ [F (ζ_{m}) : F]$ , we show that $N / F$ has a $p$ -NIB if and only if $N (ζ_{m}) / F (ζ_{m})$ has a $p$ -NIB (Theorem 1). When $p$ divides $[F (ζ_{m}) : F]$ , we show that this...

Relative Bogomolov extensions

Robert Grizzard (2015)

Acta Arithmetica

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A subfield K ⊆ ℚ̅ has the Bogomolov property if there exists a positive ε such that no non-torsion point of $K^{\times}$ has absolute logarithmic height below ε. We define a relative extension L/K to be Bogomolov if this holds for points of $L^{\times} ∖ K^{\times}$ . We construct various examples of extensions which are and are not Bogomolov. We prove a ramification criterion for this property, and use it to show that such extensions can always be constructed if some rational prime has bounded ramification index in K. ...

On the Stark-Shintani units and the ideal class groups in the cyclotomic $ℤ_{p}$ -extensions of class fields over real quadratic fields

Tsuyoshi Itoh (2002)

Acta Arithmetica

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A note on factorization of the Fermat numbers and their factors of the form $3 h 2^{n} + 1$

Michal Křížek, Jan Chleboun (1994)

Mathematica Bohemica

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We show that any factorization of any composite Fermat number $F_{m} = {2^{2}}^{m} + 1$ into two nontrivial factors can be expressed in the form $F_{m} = (k 2^{n} + 1) (ℓ 2^{n} + 1)$ for some odd $k$ and $ℓ, k \geq 3, ℓ \geq 3$ , and integer $n \geq m + 2, 3 n < 2^{m}$ . We prove that the greatest common divisor of $k$ and $ℓ$ is 1, $k + ℓ \equiv 0 m o d 2^{n}, m a x (k, ℓ) \geq F_{m - 2}$ , and either $3 | k$ or $3 | ℓ$ , i.e., $3 h 2^{m + 2} + 1 | F_{m}$ for an integer $h \geq 1$ . Factorizations of $F_{m}$ into more than two factors are investigated as well. In particular, we prove that if $F_{m} = {(k 2^{n} + 1)}^{2} (ℓ 2^{j} + 1)$ then $j = n + 1, 3 | ℓ$ and $5 | ℓ$ .

On the class numbers of real cyclotomic fields of conductor pq

Eleni Agathocleous (2014)

Acta Arithmetica

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The class numbers h⁺ of the real cyclotomic fields are very hard to compute. Methods based on discriminant bounds become useless as the conductor of the field grows, and methods employing Leopoldt's decomposition of the class number become hard to use when the field extension is not cyclic of prime power. This is why other methods have been developed, which approach the problem from different angles. In this paper we extend one of these methods that was designed for real cyclotomic fields...

On p-class groups of cyclic extensions of prime degree p of certain cyclotomic fields.

Frank Gerth (1991)

Manuscripta mathematica

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Normal integral bases and tameness conditions for Kummer extensions

Ilaria Del Corso, Lorenzo Paolo Rossi (2013)

Acta Arithmetica

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We present a detailed analysis of some properties of a general tamely ramified Kummer extension of number fields L/K. Our main achievement is a criterion for the existence of a normal integral basis for a general Kummer extension, which generalizes the existing results. Our approach also allows us to explicitly describe the Steinitz class of L/K and we get an easy criterion for this class to be trivial. In the second part of the paper we restrict to the particular case of tame Kummer...

Solution of small class number problems for cyclotomic fields

John Myron Masley (1976)

Compositio Mathematica

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