  We analyse the exact solution of the eigenproblem for the Heisenberg
Hamiltonian of magnetic heptagon, i.e. the ring of N=7 nodes, each with spin
1/2, within the XXX model with nearest neighbour interactions, from the point
of view of finite extensions of the field $\mathbb{Q}$ of rationals. We point
out, as the main result, that the associated arithmetic structure of these
extensions makes natural an introduction of some Galois qubits. They are
two-dimensional subspaces of the Hilbert space of the model, which admit a
quantum informatic interpretation as elementary memory units for a
(hypothetical) computer, based on their distinctive properties with respect to
the action of related Galois group for indecomposable factors of the secular
determinant. These Galois qubits are nested on the lattice of subfields which
involves several minimal fields for determination of eigenstates (the complex
Heisenberg field), spectrum (the real Heisenberg field), and Fourier transforms
of magnetic configurations (the cyclotomic field, based on the simple 7th root
of unity). The structure of the corresponding lattice of Galois groups is
presented in terms of Kummer theory, and its physical interpretation is
indicated in terms of appropriate permutations of eigenstates, energies, and
density matrices.
