  The modern quantum theory is based on the assumption that quantum states are
represented by elements of a complex Hilbert space. It is expected that in
future quantum theory the number field will be not postulated but derived from
more general principles. We consider the choice of the number field in quantum
theory based on a Galois field (GFQT) discussed in our previous publications.
Since any Galois field is not algebraically closed, in the general case there
is no guarantee that even a Hermitian operator necessarily has eigenvalues. We
assume that the symmetry algebra is the Galois field analog of the de Sitter
algebra so(1,4) and consider spinless irreducible representations of this
algebra. It is shown that the Galois field analog of complex numbers is the
minimal extension of the residue field modulo $p$ for which the representations
are fully decomposable.
