Pólya fields and Pólya numbers

Amandine Leriche

Displaying similar documents to “Pólya fields and Pólya numbers”

Pólya fields, Pólya groups and Pólya extensions: a question of capitulation

Amandine Leriche (2011)

Journal de Théorie des Nombres de Bordeaux

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A number field $K$ , with ring of integers $𝒪_{K}$ , is said to be a Pólya field when the $𝒪_{K}$ -algebra formed by the integer-valued polynomials on $𝒪_{K}$ admits a regular basis. It is known that such fields are characterized by the fact that some characteristic ideals are principal. Analogously to the classical embedding problem in a number field with class number one, when $K$ is not a Pólya field, we are interested in the embedding of $K$ in a Pólya field. We study here two notions which can be considered...

On $2$ -class field towers of imaginary quadratic number fields

Franz Lemmermeyer (1994)

Journal de théorie des nombres de Bordeaux

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For a number field $k$ , let $k^{1}$ denote its Hilbert $2$ -class field, and put $k^{2} = {(k^{1})}^{1}$ . We will determine all imaginary quadratic number fields $k$ such that $G = G a l (k^{2} / k)$ is abelian or metacyclic, and we will give $G$ in terms of generators and relations.

Maximal unramified extensions of imaginary quadratic number fields of small conductors, II

Ken Yamamura (2001)

Journal de théorie des nombres de Bordeaux

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In the previous paper [15], we determined the structure of the Galois groups $Gal (K_{u r} / K)$ of the maximal unramified extensions $K_{u r}$ of imaginary quadratic number fields $K$ of conductors $≦ 1000$ under the Generalized Riemann Hypothesis (GRH) except for 23 fields (these are of conductors $≧ 723$ ) and give a table of $Gal (K_{u r} / K)$ . We update the table (under GRH). For 19 exceptional fields $K$ of them, we determine $Gal (K_{u r} / K)$ . In particular, for $K = 𝐐 (\sqrt{- 856})$ , we obtain $Gal (K_{u r} / K) ≅ \tilde{S_{4}} \times C_{5} and K_{u r} = K_{4}$ , the fourth Hilbert class field of $K$ . This is the first example of a number...

Fields with non-trivial Kaplansky's radical and finite square class number

M. Kula (1981)

Acta Arithmetica

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Maximal unramified extensions of imaginary quadratic number fields of small conductors

Ken Yamamura (1997)

Journal de théorie des nombres de Bordeaux

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We determine the structures of the Galois groups Gal $(K_{u r} / K)$ of the maximal unramified extensions $K_{u r}$ of imaginary quadratic number fields $K$ of conductors $≦ 420 (≦ 719$ under the Generalized Riemann Hypothesis). For all such $K$ , $K_{u r}$ is $K$ , the Hilbert class field of $K$ , the second Hilbert class field of $K$ , or the third Hilbert class field of $K$ . The use of Odlyzko’s discriminant bounds and information on the structure of class groups obtained by using the action of Galois groups on class groups is essential. We...

The class number one problem for the dihedral and dicyclic CM-fields

Stéphane Louboutin (1999)

Colloquium Mathematicae

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We recall the determination of all the dihedral CM-fields with relative class number one, and prove that dicyclic CM-fields have relative class numbers greater than one.

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 $𝒜_{4} \times 𝒞_{2}$ (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 $1$ .

On $D_{5}$ -polynomials with integer coefficients

Yasuhiro Kishi (2009)

Annales mathématiques Blaise Pascal

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We give a family of $D_{5}$ -polynomials with integer coefficients whose splitting fields over $ℚ$ are unramified cyclic quintic extensions of quadratic fields. Our polynomials are constructed by using Fibonacci, Lucas numbers and units of certain cyclic quartic fields.