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Computing all monogeneous mixed dihedral quartic extensions of a quadratic field

István GaálGábor Nyul — 2001

Journal de théorie des nombres de Bordeaux

Let M be a given real quadratic field. We give a fast algorithm for determining all dihedral quartic fields K with mixed signature having power integral bases and containing M as a subfield. We also determine all generators of power integral bases in K . Our algorithm combines a recent result of Kable [9] with the algorithm of Gaál, Pethö and Pohst [6], [7]. To illustrate the method we performed computations for M = ( 2 ) , ( 3 ) , ( 5 ) .

Index form equations in quintic fields

István GaálKálmán 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...

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

István GaálGábor Petrányi — 2014

Czechoslovak Mathematical Journal

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 3 and t 2 + 16 has no odd square factors. In addition to generators of power integral bases we also calculate the minimal...

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