In this paper, a polar code for the m-user multiple access channel (MAC) with binary inputs is constructed. In particular, Arikan's polarization technique applied individually to each user polarizes any m-user binary input MAC into a finite collection of extremal MACs. The extremal MACs have a number of desirable properties: (i) the 'uniform sum rate'1 of the original channel is not lost, (ii) the extremal MACs have rate regions that are not only polymatroids but matroids and thus (iii) their uniform sum rate can be reached by each user transmitting either uncoded or fixed bits; in this sense they are easy to communicate over. A polar code can then be constructed with an encoding and decoding complexity of O(n log n) (where n is the block length), a block error probability of o(exp(-n1/2-ε)), and capable of achieving the uniform sum rate of any binary input MAC with arbitrary many users. An application of this polar code construction to a coding scheme for the AWGN channel is also discussed.