Displaying similar documents to “The class number one problem for the dihedral and dicyclic CM-fields”

The class number one problem for some non-abelian normal CM-fields of degree 24

F. Lemmermeyer, S. Louboutin, R. Okazaki (1999)

Journal de théorie des nombres de Bordeaux

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We determine all the non-abelian normal CM-fields of degree 24 with class number one, provided that the Galois group of their maximal real subfields is isomorphic to 𝒜 4 , the alternating group of degree 4 and order 12 . There are two such fields with Galois group 𝒜 4 × 𝒞 2 (see Theorem 14) and at most one with Galois group SL 2 ( 𝔽 3 ) (see Theorem 18); if the generalized Riemann hypothesis is true, then this last field has class number 1 .

Nonsolvable nonic number fields ramified only at one small prime

Sylla Lesseni (2006)

Journal de Théorie des Nombres de Bordeaux

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We prove that there is no primitive nonic number field ramified only at one small prime. So there is no nonic number field ramified only at one small prime and with a nonsolvable Galois group.

Pólya fields and Pólya numbers

Amandine Leriche (2010)

Actes des rencontres du CIRM

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A number field K , with ring of integers 𝒪 K , is said to be a Pólya field if the 𝒪 K -algebra formed by the integer-valued polynomials on 𝒪 K admits a regular basis. In a first part, we focus on fields with degree less than six which are Pólya fields. It is known that a field K is a Pólya field if certain characteristic ideals are principal. Analogously to the classical embedding problem, we consider the embedding of K in a Pólya field. We give a positive answer to this embedding problem by...

On the maximal unramified pro-2-extension over the cyclotomic ℤ 2 -extension of an imaginary quadratic field

Yasushi Mizusawa (2010)

Journal de Théorie des Nombres de Bordeaux

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For the cyclotomic ℤ 2 -extension k ∞ of an imaginary quadratic field k , we consider the Galois group G ( k ∞ ) of the maximal unramified pro- 2 -extension over k ∞ . In this paper, we give some families of k for which G ( k ∞ ) is a metabelian pro- 2 -group with the explicit presentation, and determine the case that G ( k ∞ ) becomes a nonabelian metacyclic pro- 2 -group. We also calculate Iwasawa theoretically the Galois groups of 2 -class field towers of certain cyclotomic 2 -extensions.

Bernoulli numbers, Hurwitz numbers, p-adic L-functions and Kummer's criterion.

Alvaro Lozano Robledo (2007)

RACSAM

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Let K = Q(ζ) and let h be its class number. Kummer showed that p divides h if and only if p divides the numerator of some Bernoulli number. In this expository note we discuss the generalizations of this type of criterion to totally real fields and quadratic imaginary fields.

PSL ( 2 , 7 ) septimic fields with a power basis

Melisa J. Lavallee, Blair K. Spearman, Qiduan Yang (2012)

Journal de Théorie des Nombres de Bordeaux

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We give an infinite set of distinct monogenic septimic fields whose normal closure has Galois group P S L ( 2 , 7 ) .