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最大公因式是多项式理论中的一个重要内容。一般的“高等代数”教材往往都局限于介绍“求最大公因式”的辗转相除法,很少论及“求最大公因式”这一代数运算的运算性质。事实上,从代数运算的角度来讨论“求最大公因式”,研究这种运算的运算性质,有助于不少问题的解决。这一点,在有关整除和互素的很多证明过程中,尤为明显。 设P为数域,f_1(x),f2(x),… ,f_n(x)∈P[x],(n≥2),当它们全为零多项式时,规定(f_1(x),f_2(x),…,f_n(x))为零多项式;当它们不全为零多项式时,规定(f_1(x),f_2(x),…,f_n(x))是当们的首系数为1的最大公因式。
The greatest common factor is an important part of polynomial theory. Ordinary “higher algebra” textbooks tend to be limited to the introduction of the “division of the greatest common divisive method”, seldom talk about “computing the greatest common divisor” algebraic computing nature of the operation. In fact, discussing “seeking the greatest common divisor” from the perspective of algebraic operations and studying the computing nature of such operations help to solve many problems. This is especially evident in the many proofs about divisibility and complementarity. Let P be the number domain, f_1 (x), f2 (x), ..., f_n (x) ∈P [x], (n≥2), when they are all zero polynomials, (f_1 (x), f_2 (x), ..., f_n (x)) are zero polynomials when they are not zero polynomials, The greatest common factor.