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

@article{Gal2014CalculatingAE,
  title={Calculating all elements of minimal index in the infinite parametric family of simplest quartic fields},
  author={Istv{\'a}n Ga{\'a}l and G. G. Petr{\'a}nyi},
  journal={Czechoslovak Mathematical Journal},
  year={2014},
  volume={64},
  pages={465-475}
}
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 familiy 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 Pt(x) = x4 − tx3 − 6x2 +tx+1, assuming that t > 0, t ≠ 3 and t2 +16 has no odd square factors. In addition to generators of… 
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References

SHOWING 1-10 OF 24 REFERENCES
Power Integral Bases in the Family of Simplest Quartic Fields
  • P. Olajos
  • Mathematics, Computer Science
    Exp. Math.
  • 2005
TLDR
This paper explicitly gives all generators of power integral bases in the ring of integers ℤ K of Kt assuming that t 2+16 is not divisible by an odd square.
EVALUATION OF THE DEDEKIND ZETA FUNCTIONS AT s = −1 OF THE SIMPLEST QUARTIC FIELDS
The simplest quartic field was first introduced by A. J. Lazarus. In this paper, we will evaluate zeta function of the simplest quartic field at s = −1. We first develop arithmetic of the simplest
Power integral bases in a parametric family of totally real cyclic quintics
TLDR
It is proved that cyclic quintic fields K n has a power integral basis only for n = -1, -2 and all generators of power integral bases are computed.
The simplest cubic fields
Abstract. The cyclic cubic fields generated by x3 = ax2 + (a + 3)x + 1 are studied in detail. The regulators are relatively small and are known at once. The class numbers are al2 2 ways of the form A
Diophantine equations and power integral bases - new computational methods
  • I. Gaál
  • Mathematics, Computer Science
  • 2002
TLDR
This work examines the latest algorithms and tools to solve classical types of diophantine equations and some infinite parametric families of fields, as well as the resolution of the corresponding infinite parametrical families of diophile equations.
Establishing the minimal index in a parametric family of bicyclic biquadratic fields
TLDR
The minimal index is found and all elements with minimal index in the bicyclic biquadratic field K = ℚ(√(4c + 1) c, √(c − 1)c).
Simultaneous Representation of Integers by a Pair of Ternary Quadratic Forms—With an Application to Index Form Equations in Quartic Number Fields
LetQ1, Q2∈Z[X, Y, Z] be two ternary quadratic forms andu1, u2∈Z. In this paper we consider the problem of solving the system of equations[formula]According to Mordell [12] the coprime solutions
A parametric family of quintic Thue equations
TLDR
This paper investigates the family of Thue equations F(x,y) originating from Emma Lehmer's family of quintic fields, and shows that for |t| > 3.28.10 15 the only solutions are the trivial ones with x= 0 or y = 0.
Solving index form equations in the two parametric families of biquadratic fields
In this paper we nd a minimal index and determine all integral elements with the minimal index in two families of totally real bicyclic biquadratic elds
A Parametric Family of Quintic Thue Equations II
Abstract. In this paper we completely solve the family of Thue equations where is an integral parameter. In particular, for , the only solutions are the trivial ones with x = 0 or y = 0. The result
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