  It has been shown that an extension of the basic binary polar transformation
also polarizes over finite fields. With it the direct encoding of q-ary sources
and channels is a process that can be implemented with simple and efficient
algorithms. However, direct polar decoding of q-ary sources and channels is
more involved. In this paper we obtain a recursive equation for the likelihood
ratio expressed as a LR vector. With it successive cancellation (SC) decoding
is applied in a straightforward way. The complexity is quadratic in the order
of the field, but the use of the LR vector introduces factors that soften that
complexity. We also show that operations can be parallelized in the decoder.
The Bhattacharyya parameters are expressed as a function of the LR vectors, as
in the binary case, simplifying the construction of the codes. We have applied
direct polar coding to several sources and channels and we have compared it
with other multilevel strategies. The direct q-ary polar coding is closer to
the theoretical limit than other techniques when the alphabet size is large.
Our results suggest that direct q-ary polar coding could be used in real
scenarios.
