Integer Coefficients Polynomial Factorization Method Study

The factorization problem of the whole coefficient polynomial can be divided into two types, one is to study the irreducible problem, and the other is to study the reducible problem. And in reducible condition, it is necessary to continue to study how to carry out factorization.Eisenstin judging method is to judge whether the whole coefficient polynomial is irreducible. There are not many whole coefficient polynomials, in which Eisenstin judging method can be directly applied. However, with the proper transformation of the whole coefficient polynomial and promotion of the Eisenstin judging method, the application scope of Eisenstin judging method has been expanded.In the complex number field, it is common to study the factorization of polynomial by means of root. Apply this idea in the rational field, research into the factorization of the whole coefficient polynomial by original root, and with its value, we can study the irreducibility of the whole coefficient polynomial.In theory, Kronecker method has resolved the reducibility problem of the whole coefficient polynomial in rational field. In other words, the whole coefficient polynomial f(x) =a_nxⁿ + a_n-1x^n-1+…+a₁x+a₀can always be decomposed into the products of irreducible factors by limited steps of factorization in rational field.Kronecker method is only feasible in theory, and it is difficult to be applied in the actual operation of factorization, thus it lacks practicality. Theories with practical value are more meaningful, in order to achieve the practical value of Kronecker method, and inherit the fine traditions of China's ancient mathematics as well. We make use of modern computer's characteristic in dealing with a great deal of data in a short period of time, and the procedure design to solve the problem that all the whole coefficient polynomial f(x) =a_nxⁿ + a_n-1x^n-1+…+a₁x+a₀can be decomposed into the products of irreducible factors in the rational field. It fills the gaps in mathematical operation.