We present a method for factorizing multivariate polynomials over algebraic fields obtained from successive extensions of the field of rational numbers. The basic idea underlying this method is the reduction of polynomial factorization over algebraic extension fields to the factorization over the rational number field via linear transformation and the computation of characteristic sets with respect to a proper variable ordering. The factors over the algebraic extension fields are finally determined via greatest-common-divisor computation. This method has been implemented in the Maple system. Preliminary experiments show that it is rather efficient. We give timing statistics in Maple 4.3 on 40 test examples taken from the literature or randomly generated. For all those examples to which the Maple built-in algorithm is applicable, our algorithm is always faster.