形式导数

前置

• 域论(域拓展)

定义

• Let $F$ be an arbitrary field. We define the derivative(导数、形式导数) of $p(x)={\sum }_{k=0}^n{a}_k{x}^k\in F[x]$ to be $D(p(x))={\sum }_{k=1}^{n}k{a}_k{x}^{k-1}$.

性质

16. Let $p(x),q(x)\in F[x]$, and let $n$ and $m$ be nonnegative integers. The following statements hold:
    • a. $D(p(x)+q(x))=D(p(x))+D(q(x))$.
    • b. $D(ap(x))=aD(p(x))$ for any $a\in F$.
    • c. $D(x^n{x}^m)=x^{n}D(x^m)+D(x^n)x^m$.
    • d. $D(p(x)x^m)=p(x)D(x^m)+D(p(x))x^m$.
    • e. $D(p(x)q(x))=p(x)D(q(x))+D(p(x))q(x)$.
    • f. $F\le E,a\in E$，$(x-a{)}^2$ divides $p(x)$ if and only if $a$ is a zero of both $p(x)$ and $D(p(x))$.