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Index for subgroups of the group of units in number fields

Tsutomu Shimada (1998)

Acta Arithmetica

We define a sequence of rational integers u i ( E ) for each finite index subgroup E of the group of units in some finite Galois number fields K in which prime p ramifies. For two subgroups E’ ⊂ E of finite index in the group of units of K we prove the formula v p ( [ E : E ' ] ) = i = 1 r u i ( E ' ) - u i ( E ) . This is a generalization of results of P. Dénes [3], [4] and F. Kurihara [5].

Index form equations in quintic fields

István Gaál, Kálmán Győry (1999)

Acta Arithmetica

The problem of determining power integral bases in algebraic number fields is equivalent to solving the corresponding index form equations. As is known (cf. Győry [25]), every index form equation can be reduced to an equation system consisting of unit equations in two variables over the normal closure of the original field. However, the unit rank of the normal closure is usually too large for practical use. In a recent paper Győry [27] succeeded in reducing index form equations to systems of unit...

Nombre de classes et unités des corps de nombres cycliques quintiques d'E. Lehmer

Stéphane Jeannin (1996)

Journal de théorie des nombres de Bordeaux

Le but de cet article est l’étude des corps cycliques quintiques définis par les polynômes d’E. Lehmer. On calcule premièrement le conducteur de ces corps dans le cas général (non nécessairement premier) puis on généralise un théorème (qui donne les unités de ces corps) démontré par R. Schoof et L.C. Washington. Par la méthode de dévissage des unités cyclotomiques, qui calcule le nombre de classes et les unités, on dresse une table de ces corps particuliers (de conducteur f 3000000 ) et de leur nombre de...

On monogenity of certain pure number fields of degrees 2 r · 3 k · 7 s

Hamid Ben Yakkou, Jalal Didi (2024)

Mathematica Bohemica

Let K = ( α ) be a pure number field generated by a complex root α of a monic irreducible polynomial F ( x ) = x 2 r · 3 k · 7 s - m [ x ] , where r , k , s are three positive natural integers. The purpose of this paper is to study the monogenity of K . Our results are illustrated by some examples.

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