Simplest quartic and simplest sextic Thue equations over imaginary quadratic fields

Abstract

The families of simplest cubic, simplest quartic and simplest sextic fields and the related Thue equations are well known, see [18], [17]. The family of simplest cubic Thue equations was already studied in the relative case, over imaginary quadratic fields. In the present paper we give a similar extension of simplest quartic and simplest sextic Thue equations over imaginary quadratic fields. We explicitly give the solutions of these infinite parametric families of Thue equations over arbitrary imaginary quadratic fields. new national excellence program of the ministry of human capacities. are up to sign given by the following: for any m and any t: (x, y) = (0, 0), (0, 1), (1, 0), for any m and any t = 1: (x, y) = (1, 2), (2, −1), for any m and any t = −1: (x, y) = (2, 1), (−1, 2), for any m and any t = 4: (x, y) = (2, 3), (3, −2), for any m and any t = −4: (x, y) = (3, 2), (−2, 3), for m = 1 and any t: (x, y) = (0, i), (i, 0), for m = 3 and any t: (x, y) = (ω, 0), (0, ω), (1 − ω, 0), (0, 1 − ω),

A generalization of simplest number fields and their integral basis

“…However these period lengths are much smaller than we give in Theorem 16, the smallest period lengths are even smaller. It was shown in [8], that an integral basis of the simplest sextic fields is repeating periodically modulo 36 instead of 54 6 . In order to find the smallest period lengths, we have to calculate an integral basis of the fields belong to the parameters less than the period length given in the table above, and detect the h (r) 0 , .…”

“…(X) and r (n−1) (X). However, the determinant of this matrix is α 2 + α + 1, which is not zero, since α is a root of r (n) (X), and by (6), the cube roots of unity are not roots of r (n) (X). Iteratively using this correspondence, we get, that α is a common root of f (1) m (X) and r (1) (X), but r (1) (X) = −1 does not have any root, which is contradiction.…”

“…E.Thomas [28], M.Mignotte [22], G.Lettl, A.Pethő and P.Voutier [21], [20] and G.Lettl and A.Pethő [19], I.Gaál [5] solved Thue equations corresponding to the simplest polynomials in absolute case, and C.Heuberger [11], I.Gaál, B.Jadrijević and L.Remete [6] in certain relative cases. A.Hoshi [12], [13], [14] gave a correspondence between solutions of a family of Thue equations and the isomorphism classes of the simplest number fields.…”

“…where a = 0 or −1 for degree 3, and a = 1 or −2 for degree 6, then we can write the minimal polynomial of β over Q in the form: 6) m (X) = X 6 − 6mX 5 − 15(m + 1)X 4 − 20X 3 + 15mX 2 + 6(m + 1)X + 1. Therefore, if m is a rational parameter, then the polynomials above are cyclic, they have rational coefficients and σ : z → az − 1 z + a + 1 permutes their roots transitively.…”

Thue equations over $\mathbb{C}(T)$: The Complete Solution of a Simple quartic family

“…Such a conjecture was eventually proved correct by Mignotte [14]. More general questions related to such Thue equations were addressed in [6,10]. Lettl and Pethő [9] then investigated the family of quartic forms (2)…”