  We study the complexity of polynomial multiplication over arbitrary fields.
We present a unified approach that generalizes all known asymptotically fastest
algorithms for this problem. In particular, the well-known algorithm for
multiplication of polynomials over fields supporting DFTs of large smooth
orders, Sch\"onhage-Strassen's algorithm over arbitrary fields of
characteristic different from 2, Sch\"onhage's algorithm over fields of
characteristic 2, and Cantor-Kaltofen's algorithm over arbitrary algebras---all
appear to be instances of this approach. We also obtain faster algorithms for
polynomial multiplication over certain fields which do not support DFTs of
large smooth orders.
  We prove that the Sch\"onhage-Strassen's upper bound cannot be improved
further over the field of rational numbers if we consider only algorithms based
on consecutive applications of DFT, as all known fastest algorithms are. We
also explore the ways to transfer the recent F\"urer's algorithm for integer
multiplication to the problem of polynomial multiplication over arbitrary
fields of positive characteristic.
  This work is inspired by the recent improvement for the closely related
problem of complexity of integer multiplication by F\"urer and its consequent
modular arithmetic treatment due to De, Kurur, Saha, and Saptharishi. We
explore the barriers in transferring the techniques for solutions of one
problem to a solution of the other.
