Algebraic function fields with equal class number

James Leitzel; Manohar Madan

Displaying similar documents to “Algebraic function fields with equal class number”

Algebraic function fields of class number one

Manohar Madan, Clifford Queen (1972)

Acta Arithmetica

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On the genus group of algebraic number fields

Kiyoaki Iimura (1984)

Acta Arithmetica

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Artin's primitive root conjecture for quadratic fields

Hans Roskam (2002)

Journal de théorie des nombres de Bordeaux

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Fix an element $α$ in a quadratic field $K$ . Define $S$ as the set of rational primes $p$ , for which $α$ has maximal order modulo $p$ . Under the assumption of the generalized Riemann hypothesis, we show that $S$ has a density. Moreover, we give necessary and sufficient conditions for the density of $S$ to be positive.

Le Veque's superelliptic equation over function fields

R. Mason, B. Brindza (1986)

Acta Arithmetica

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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...

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...

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...

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

M. Kula (1981)

Acta Arithmetica

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On the representation of units by cyclotomic polynomials

Veikko Ennola (1980)

Acta Arithmetica

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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 $1$ .