Displaying similar documents to “Power integral bases in the family of simplest quartic fields.”

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

Computing all monogeneous mixed dihedral quartic extensions of a quadratic field

István Gaál, Gábor Nyul (2001)

Journal de théorie des nombres de Bordeaux

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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 = ℚ ( 2 ) , ℚ ( 3 ) , ℚ ( 5 ) .