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 $O(2^{-N^{0.5 - ε}})$ is guaranteed. Examples of achievable rates for the $3$-user multiple access channel are provided.