In this paper, polar codes for the m-user multiple access channel (MAC) with binary inputs are constructed. It is shown that Arikan's polarization technique applied individually to each user transforms independent uses of an m-user binary input MAC into successive uses of extremal MACs. This transformation has a number of desirable properties: 1) the "uniform sum-rate" of the original MAC is preserved, 2) the extremal MACs have uniform rate regions that are not only polymatroids but matroids, and thus, 3) 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 is the block length), a block error probability o(exp(-n(1/2-epsilon))) of, and capable of achieving the uniform sum-rate of any binary input MAC with arbitrary many users. Applications of this polar code construction to channels with a finite field input alphabet and to the additive white Gaussian noise channel are also discussed.