On correspondence between solutions of a family of cubic Thue equations and isomorphism classes of the simplest cubic fields

Abstract

Let m ≥ −1 be an integer. We give a correspondence between integer solutions to the parametric family of cubic Thue equationswhere λ > 0 is a divisor of m 2 + 3m + 9 and isomorphism classes of the simplest cubic fields. By the correspondence and R. Okazaki's result, we determine the exactly 66 non-trivial solutions to the Thue equations for positive divisors λ of m 2 + 3m + 9. As a consequence, we obtain another proof of Okazaki's theorem which asserts that the simplest cubic fields are non-isomorphic to each other except for m

Complete solutions to a family of Thue equations of degree 12

“…The aim of this paper is to generalize the results in [15,16] The polynomial f m (X) = F m (X, 1) is irreducible over Q if m ∈ Z \ {−8, −3, 0, 5}. In general, however, the root field Q(θ) with f m (θ) = 0 is not a Galois extension of Q.…”

“…A family of Thue equations of degree 8 is solved by Heuberger, Togbé and Ziegler [14]. In [15] and [16], the author determined solutions to the families of Thue equations F (d) m (X, Y ) = λ d of degree d = 3 and 6 where m ∈ Z, λ 3 is a divisor of m 3 + 3m + 9 and λ 6 is a divisor of 27(m 2 + 3m + 9). See also the quartic case [17].…”

“…We use Okazaki's theorem (see [15,Theorem 1.4]) which claims that for m ≥ −1, the simplest cubic fields are non-isomorphic to each other except for m = −1, 0, 1, 2, 3, 5, 12, 54, 66, 1259, 2389. Okazaki's theorem was reproved in [15,Section 1].…”

“…The method of this paper, the field isomorphism method, is developed in [18], [15] (see also [22]) and applied in [16] and [4]. It uses the splitting…”

On cubic Thue equations and the indices of algebraic integers in cubic fields

“…In 2011, A. Hoshi [15] studied the case when k is a positive divisor of n 2 + 3n + 9, and gave a correspondence between integer solutions to the parametric family of cubic Thue equations…”