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两个多項式的最大公因式通常都是經过輾轉相除而求得,这种运算既煩瑣又容易出差錯,本文介紹一个新的簡易的求法。引理工Ⅰ体p上多項式f(x),g(x)的最大公因式和f(x),cg(x)的最大公因式相同,其中c是体P中任一非零元。引理Ⅱ体P上多項式f(x),g(x)的最大公因式和f(x),f(x)+g(x)的最大公因式相同。引理Ⅲ如果不考虑因式x~(?)(这种因式徂容易判断,以下称为显然因式),则体P上多項式f(x),g(x)的最大公因式与f(x),xg(x)的最大公因式相同。在下列討論中,把多項式f(x)=ax~n+a_1x~(n-1)++…+a_n-1~x+a_n和g(x)=b_0x~n+b_1x~(n-1)+…++b_(n-1)x+b_n的最大公因式記成矩陣:
The largest common factor of two polynomials is usually obtained by diversion. This kind of operation is both tedious and error-prone. This paper introduces a new simple solution. The maximum common factor of the polynomial f(x), g(x) on the polynomial I is the same as the maximum common factor of f(x) and cg(x), where c is any non-zero element in the volume P. The maximal common factor of the polynomial f(x),g(x) on the lemma II P is the same as the maximum common factor of f(x),f(x)+g(x). Lemma III If we do not consider the factor x ~ (?) (this factor is easy to determine, the following is called the obvious factor), then the largest common factor of the polynomial f(x),g(x) on the body P The greatest common factor of f(x),xg(x) is the same. In the following discussion, the polynomials f(x)=ax~n+a_1x~(n-1)++...+a_n-1~x+a_n and g(x)=b_0x~n+b_1x~(n-1) ) +... ++b_(n-1)x+b_n The largest common factor score matrix: