Bethe Ansatz, Galois symmetries, and finite quantum systems
We demonstrate some applications of arithmetic structures in Bethe Ansatz — the famous substitution which yields an exact solution of a quantum $N$-body problem for the linear magnetic ring of $N$ spins $1/2$ within the XXX model. We point out the purely arithmetic form of the eigenproblem of the associated Heisenberg Hamiltonian in the initial (calculational) basis of all magnetic configurations, together with the resulting solution, expressed in terms of a finite extension of the prime field of rationals. The Galois group of this extension acquires the natural physical interpretation in terms of admissible permutations of rigged string configurations. The cyclotomic number field proves to be an important subfield of this extension, responsible for the translational symmetry of the magnetic ring, reflected in quasimomenta of a finite Brilloin zone. We also point out the role of the cyclotomic fields in determination of all mutually unbiased bases for a Hilbert space with the dimension $N$ being a power of a prime.