Galois symmetry of string structure of quanta of phase in the XXX heptagonal ring

An arithmetic qubit, separated from the state space of the heptagonal magnetic ring within the XXX model in the three-magnon sector at the center of the Brillouin zone, is shown to have a Galois symmetry, stemming from a polynomial f of degree 6 indecomposable over the prime field Q, with admissible quanta of phase as its roots. The Galois group is shown to be isomorphic with the dihedral group D6, and the main Galois theorem for this case is demonstrated in detail, including generators for each subfield, the corresponding minimal polynomials, as well as intepretation in terms of rigged string configurations. The splitting number field for the polynomial f proves to be the arena for exact solutions of Bethe Ansatz equations for this case.