Computing all monogeneous mixed dihedral quartic extensions of a quadratic field

István Gaál; Gábor Nyul

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Displaying similar documents to “Computing all monogeneous mixed dihedral quartic extensions of a quadratic field”

Calculating all elements of minimal index in the infinite parametric family of simplest quartic fields

István Gaál, Gábor Petrányi (2014)

Czechoslovak Mathematical Journal

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It is a classical problem in algebraic number theory to decide if a number field is monogeneous, that is if it admits power integral bases. It is especially interesting to consider this question in an infinite parametric family of number fields. In this paper we consider the infinite parametric family of simplest quartic fields $K$ generated by a root $ξ$ of the polynomial $P_{t} (x) = x^{4} - t x^{3} - 6 x^{2} + t x + 1$ , assuming that $t > 0$ , $t \neq 3$ and $t^{2} + 16$ has no odd square factors. In addition to generators of power integral bases we also calculate...

Power integral bases in the family of simplest quartic fields.

Olajos, Péter (2005)

Experimental Mathematics

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Non-monogenity of multiquadratic number fields

Gábor Nyul (2002)

Acta Mathematica et Informatica Universitatis Ostraviensis

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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...

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

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...