Factorization of polynomials is a strong computational tool in algebraic geometry and computer algebra. This plays an important role in industry and Mathematics (especially because of its application in solving polynomial systems). In 1967, Berlekamp [3] has presented an algorithm for he factorization of polynomials over finite fields. He has used the invariant space of Frobenius's map over the quotient ring. This method is very important, because using the factorization over finite fields, we can factor the polynomials in the more complicated polynomial rings like $\ Z[x]$, \ [x_1,\ldots,x_n]$ and $\ {GF}_q[x_1,\ldots,x_n]$, where $\ {GF}_q$ denotes a finite field of q$ elements. Then, Zassenhaus [48] has used Berlekamp' method and Hensel's lemma to propose a Few method for factorization of univariate polynomials over integers. This algorithm first reduces the factorization of a given univariate polynomial to the factorization of the polynomial over a finite field $\ F_p$ for a convenient prime number $p$. After factoring this polynomial over $\ F_p$ by Berlekamp's algorithm, we can lift the factorization to $\ F_{p^2}, \ldots,\ F_{p^N}$ by Hensel's lemma for $N$ enough large. In 1975, using the same approach, Wang and Rotchild [46] have described a new algorithm for factoring the multivariate polynomials over integers. This algorithm begins by substituting all the variables (except one of them) by some selected integers, and obtains a univariate polynomial. Then, we use Zassenhaus method to factorize this new polynomial. Finally, we lift this factorization to the factorization of the give multivariate polynomial by a generelization of Hensel's lemma.\\ In the rest of this thesis, we deal with the concept of primary decomposition and its relation to factorization. In this direction, in 2002, Monico [37] has presented an applicable algorithm for computing the primary decomposition of zero dimensional ideals over infinite fields. It is worth noting that zero dimensional ideals are the most important ideals in practice, because the corresponding system has a finite number of solutions and they have many applications in industry. In 2009, Gao et al [18] has described a new algorithm by using Berlekamp's algorithm to compute a primary decomposition of a zero dimensional fields. All the algorithm described in this thesis have been implemented in Maple.ideal over finite