将带整系数或有理系数的多項式分解为带整系数或有理系数的不可約多項式的乘积,是中学里因式分解教学中的主要問題,也是一般中学师生感到困难的問題。困难主要是两个“心中无数”:第一,是否已經分到不能再分,心中无数;第二,分解方法是否合理,心中无数。其实对于这两“无数”,下面的克郎湼克定理可以完全解决的。克郎湼克定理。設f(x)是任意一个带有理系数的次数≥1的多項式,那末經过有限次有理运算之后,永远可以将f(x)分解为一些带有理系数的不可約多項式的乘积。单从定理的陈述来看,只能說这个定理肯定了整数系数和有理系数多項式因式分解的可能性,但这个定理的証明过程,也給出了找f(x)的不可約因式的具体途径。所以說这个定理能够解决上述的两个問題。可是这个定理在高等学校代数教科书中很少提到。原 Decomposing polynomials with integral coefficients or rational coefficients into an irreducible polynomial product with integral coefficients or rational coefficients is a major problem in the middle-school factorization education, and it is also a problem for teachers and students in general middle schools. The difficulties are mainly two “numerous hearts”: first, whether it has been assigned to can not be divided, countless hearts; second, the decomposition method is reasonable, countless hearts. In fact, for these two “numerous”, the following Kroneck theorem can be completely resolved. Kroneck Theorem. Let f(x) be any polynomial of degree ≥ 1 with rational coefficients, then after a finite number of rational operations, we can always decompose f(x) into some irreducible polynomial products with rational coefficients. From the statement of the theorem alone, we can only say that this theorem affirms the possibility of factorization of integer and rational polynomials, but the proof process of this theorem also gives the irreducible factor for f(x). Specific ways. So this theorem can solve the above two problems. However, this theorem is rarely mentioned in algebra textbooks in colleges and universities. original