OFFSET
0,1
COMMENTS
For any totally real field K, an additively indecomposable element of K is a totally positive element in the maximal order of K which cannot be written as the sum of two totally positive integral elements of K. Here, an element x of K is totally positive if all conjugates of x are positive real numbers.
Let K be a simplest cubic field with defining polynomial f(x) = x^3 - n*x^2 - (n+3)*x - 1, and let rho be a root of f(x). Kala and Tinková classified all additively indecomposable elements of K in the case where the ring of integers O_K is Z[rho], and Gil-Muñoz and Tinková classified all additively indecomposable elements of K in the case where Z[rho] has index 3 in O_K.
LINKS
Robin Visser, Table of n, a(n) for n = 0..240
Daniel Gil-Muñoz and Magdaléna Tinková, Additive structure of non-monogenic simplest cubic fields, Ramanujan J. 66 (2025), no. 3, Paper No. 47, 56 pp.
Vítězslav Kala, Universal quadratic forms and indecomposables in number fields: a survey, Commun. Math. 31 (2023), no. 2, 81-114.
Vítězslav Kala and Magdaléna Tinková, Universal quadratic forms, small norms, and traces in families of number fields. Int. Math. Res. Not. IMRN 2023, no. 9, 7541-7577.
D. Shanks, The simplest cubic fields, Math. Comp., 28 (1974), 1137-1152.
FORMULA
a(n) = (n^2 + 3*n + 6)/2 = A152948(n+3) if the maximal order of the field Q[x]/(x^3 - n*x^2 - (n+3)*x - 1) is generated by a root of x^3 - n*x^2 - (n+3)*x - 1 [Kala-Tinková, Section 7.2].
a(n) = (n^2 + 39*n + 36)/18 if n > 3 and if the order Z[rho] generated by a root rho of x^3 - n*x^2 - (n+3)*x - 1 has index 3 in the maximal order of Q[x]/(x^3 - n*x^2 - (n+3)*x - 1) [Gil-Muñoz-Tinková, Theorem 1.1].
EXAMPLE
For n = 0, the corresponding simplest cubic field is given by the defining polynomial f(x) = x^3 - 3*x - 1. All additively indecomposable elements of K are 1, -rho^2 + 4, and rho^2 + rho + 1 (up to multiplication by totally positive units), thus a(0) = 3.
For n = 1, the corresponding simplest cubic field is given by the defining polynomial f(x) = x^3 - x^2 - 4*x - 1. All additively indecomposables elements of K are 1, rho^2 + rho + 1, rho^2 - 2*rho, -rho^2 + rho + 5, and rho + 2 (up to multiplication by totally positive units), thus a(1) = 5.
For n = 2, the corresponding simplest cubic field is given by the defining polynomial f(x) = x^3 - 2*x^2 - 5*x - 1. All additively indecomposables elements of K are 1, rho^2 - 3*rho, -rho^2 + 2*rho + 6, rho + 2, 2*rho^2 - 5*rho - 1, rho^2 - 2*rho, 2*rho^2 - 6*rho - 1, and -4*rho^2 + 9*rho + 19 (up to multiplication by totally positive units), thus a(2) = 8.
CROSSREFS
KEYWORD
nonn
AUTHOR
Robin Visser, Sep 02 2025
STATUS
approved