  Quantization is studied from a viewpoint of field extension. If the dynamical
fields and their action have a periodicity, the space of wave functions should
be algebraically extended `a la Galois, so that it may be consistent with the
periodicity. This was pointed out by Y. Nambu three decades ago. Having chosen
quantum mechanics (one dimensional field theory), this paper shows that a
different Galois extension gives a different quantization scheme. A new scheme
of quantization appears when the invariance under Galois group is imposed as a
physical state condition. Then, the normalization condition appears as a sum
over the product of more than three wave functions, each of which is given for
a different root adjoined by the field extension.
