  We consider the problem of polar coding for transmission over $m$-user
multiple access channels. In the proposed scheme, all users encode their
messages using a polar encoder, while a multi-user successive cancellation
decoder is deployed at the receiver. The encoding is done separately across the
users and is independent of the target achievable rate. For the code
construction, the positions of information bits and frozen bits for each of the
users are decided jointly. This is done by treating the polar transformations
across all the $m$ users as a single polar transformation with a certain
\emph{polarization base}. We characterize the resolution of achievable rates on
the dominant face of the uniform rate region in terms of the number of users
$m$ and the length of the polarization base $L$. In particular, we prove that
for any target rate on the dominant face, there exists an achievable rate, also
on the dominant face, within the distance at most $\frac{(m-1)\sqrt{m}}{L}$
from the target rate. We then prove that the proposed MAC polar coding scheme
achieves the whole uniform rate region with fine enough resolution by changing
the decoding order in the multi-user successive cancellation decoder, as $L$
and the code block length $N$ grow large. The encoding and decoding
complexities are $O(N \log N)$ and the asymptotic block error probability of
$O(2^{-N^{0.5 - \epsilon}})$ is guaranteed. Examples of achievable rates for
the $3$-user multiple access channel are provided.
