  Polar codes are constructed for m-user multiple access channels (MAC) whose
input alphabet size is a prime number. The block error probability under
successive cancelation decoding decays exponentially with the square root of
the block length. Although the sum capacity is achieved by this coding scheme,
some points in the symmetric capacity region may not be achieved. In the case
where the channel is a combination of linear channels, we provide a necessary
and sufficient condition characterizing the channels whose symmetric capacity
region is preserved upon the polarization process. We also provide a sufficient
condition for having a total loss in the dominant face.
