  In this work we obtain the exact solution for relativistic Landau problem
plus oscillator potential in a complex symmetric gauge field in a
non-commutative complex space, using the algebraic techniques of creation and
annihilation operators. It is shown that the relativistic Landau problem in a
complex symmetric gauge field in a non-commutative complex space is similar
behavior to the Pauli equation in the presence of a symmetric gauge field in a
commutative ordinary space. We derive the exact non-commutative Landau and
oscillator energy levels, while the non-relativistic limit of the energy
spectrum is obtained. We show that the energy is not degenerate and is splitted
into two levels, as in the Zeeman effect.
