Existing literature on source coding with side information (SCSI) mostly uses the state-of-the-art channel codes namely LDPC codes, turbo codes, and their variants and assume classical unique decoding. In this paper, we present an algebraic approach to SCSI based on the list decoding of the underlying channel codes. We show that the theoretical limit of SCSI can be achieved in the proposed list decoding based framework when the correlation between the source and side information is -ary symmetric. We argue that, as opposed to channel coding, the correct sequence from the list produced by the list decoder can effectively be recovered in case of SCSI with a few CRC symbols. The CRC symbols, which allow the decoder to identify the correct sequence, incur negligible overhead for large block lengths. More importantly, these CRC symbols are not subject to noise since we are dealing with a virtual noisy channel rather than a real noisy channel. Finally, we present a guideline for designing constructive SCSI schemes for non-binary and binary sources using Reed Solomon codes and BCH codes, respectively. This guideline allows us to design a SCSI scheme for any arbitrary -ary symmetric correlation without resorting to simulation.