No / A finite quantum system in which the position and momentum take values in the Galois field GF(p¿l) is constructed from a smaller quantum system in which the position and momentum take values in Zp , using field extension. The Galois trace is used in the definition of the Fourier transform. The Heisenberg¿Weyl group of displacements and the Sp(2, GF(p¿l)) group of symplectic transformations are studied. A class of transformations inspired by the Frobenius maps in Galois fields is introduced. The relationship of this 'Galois quantum system' with its subsystems in which the position and momentum take values in subfields of GF(p¿l) is discussed.