利用代数结构寻找二次根式的有理化因式

Utilize Algebra Structure to Search for the Square Radical's Rationalization Factor

考虑下面代数式a0 +a1 b1 +a2 b2 +… +an bn ( ) 的有理化因式问题 .本文区别于传统作法—利用观察、猜测的方法来求有理化因式 ,运用近世代数的有关理论 ,将形如 ( )式纳入代数系统讨论 .首先 ,将 ( )式作为有理数域的代数扩域的元素加以考虑 ,通过扩域的相关性质的分析得到它的有理化因式的一般结构 ;进而 ,运用待定系数法把求解 ( )式的有理化因式问题转化为一个关于结构系数的齐次线性方程组的求解问题 ;最后指出 ,这种方法可用计算机实现 .

Consider the rationalization factor of the folloving algebra expressima 0+a 1b 1+a 2b 2+...+a nb n (*)This articl is different from the traditional method-utilize observation and speculation to seek for rationalization factor. Then put the algebraic expression(*)into Algebra system and discuss it by using the relevant theories. First of all, this articl considers the algebraic expression(*)as one element of the algebraic extension of rational number field , then achieve the general structure of its rationalization factor according to the analysis of the relevant characteristics of algebratic extension; Second ,this articl uses the method of indeterminate coefficients to change this question into the question about the linear homogeheous equations of coefficient; At last ,the method can be gotten by using computers.