兩個本原多項式的乘積仍為本原多項式.
證明:我先舉一個例子來說明這個命題的正确性.設$a_1x+a_0$和$b_1x+b_0$都是本原多項式.
\begin{align*}
(a_1x+a_0)(b_1x+b_0)=a_1b_1x^2+(a_1b_0+a_0b_1)x+a_0b_0
\end{align*}
假設得到的
a_1b_1x^2+(a_1b_0+a_0b_1)x+a_0b_0
不是本原多項式,則
a_1b_1,a_1b_0+a_0b_1,a_0b_0
有不為$\pm 1$的公因數$s$.于是我們有
s|a_1b_1
鑒于$a,b$地位的平等,我們不妨針對$a$.不妨設存在整數$k_1(k_1\neq \pm 1)$,使得$k_1$是$a_1$和$s$的公因數.易得必有
k_1|a_0b_1
顯然,$k_1$不能是$a_0$的因數(為什麼?因為$a_0$和$a_1$互素),是以$k_1$隻能是$b_1$的因數,但是
k_1|a_0b_0
可見,$k_1$是$b_0$的因數,這與$b_0,b_1$互素沖突.
我們再來分析一個例子:若$a_2x^2+a_1x+a_0$和$b_2x^2+b_1x+b_0$都是本原多項式,則
(a_2x^2+a_1x+a_0)(b_2x^2+b_1x+b_0)
是本原多項式.
(a_2x^2+a_1x+a_0)(b_2x^2+b_1x+b_0)=a_2b_2x^4+(a_2b_1+a_1b_2)x^3+(a_2b_0+a_1b_1+a_0b_2)x^2+(a_1b_0+a_0b_1)x+a_0b_0
為此我們畫樹狀圖幫助分析.如圖,樹狀圖的每個分支結束之時,也就是推出沖突之時.
![](https://img.laitimes.com/img/9ZDMuAjOiMmIsIjOiQnIsISPrdEZwZ1Rh5WNXp1bwNjW1ZUba9VZwlHdsATOfd3bkFGazxCMx8VesATMfhHLlN3XnxCMwEzX0xiRGZkRGZ0Xy9GbvNGLpZTY1EmMZVDUSFTU4VFRR9Fd4VGdsYTMfVmepNHLrJXYtJXZ0F2dvwVZnFWbp1zczV2YvJHctM3cv1Ce-cmbw5iYmZGZmRjZwMmZzYzNiRTM4YjY4Y2NiZDM2QDN4ETY38CX2AzLchDMxIDMy8CXn9Gbi9CXzV2Zh1WavwVbvNmLvR3YxUjL4M3Lc9CX6MHc0RHaiojIsJye.png)
統統以推出沖突結束.現在我們來找出一般情形下樹狀圖的規律.對于
(a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0)(b_nx^n+b_{n-1}x^{n-1}+\cdots+b_1x+b_0)=\sum_{i=0}^{2n}\sum_{p+q=i}a_pb_qx^i(0\leq
p,q\leq n)
的樹狀圖,其實是有規律的.為了說明白這些規律,我先定義一些名詞.對于一株樹的一部分,如圖,
View Code
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我們把如圖這株樹叫三層樹,因為包含了三代.而且,我們把$p\to m,p\to a\to b,p\to a\to c$都叫做這株樹的一個枝丫.
對于
形成的樹來說,容易驗證有以下規律:
1.該株樹裡任意一個枝丫裡的同一類型的字母(同$a$或同$b$)的下标是遞降的.
2.該樹是$2n+1$層樹.
3.$a_0$或$b_0$的出現标志着沖突的出現,也就是一株樹的終結.而且每個枝丫必定以$a_0$或$b_0$終結.
現在,
所對應的樹的一個枝丫裡最多有$2n+1$個字母,而$a_i(1\leq i\leq n)$有$n$個,$b_i$也有$n$個,是以即使是字母最多的枝丫,還得留下一個空位給$a_0$或$b_0$,無論是填$a_0$還是$b_0$,都是以沖突的形式出現.而其它比$2n+1$枝丫短的枝丫早早就以$a_0$結尾而結束了.
附:多項式乘法中展現出來的遞推規律
以多項式$a_2x^2+a_1x+a_0$與$b_2x^2+b_1x+b_0$相乘為例:
我們發現,相乘得到的結果呈現出一種對稱的形式:$x^4$項的系數和常數項的系數呈現出一樣的形式,隻不過$x^4$的系數下标大$2=\frac{4-0}{2}$.$x^3$的系數和$x$的系數形式也是一樣的,隻不過$x^3$的系數下标比$x$的系數下标大$\frac{3-1}{2}$.$x^2$的系數在中間.有了這個規律,我們隻要知道常數項系數,$x$項系數,$x^2$項系數,就知道了乘出來的多項式的狀況.下面我們看系數
a_0b_0,a_1b_0+a_0b_1,a_2b_0+a_1b_1+a_0b_2
這三個系數裡蘊含了一種遞推關系.當我們令$a_0=0$,再将$a_1,a_2$的下标減去1時,變為
0,a_0b_0,a_1b_0+a_0b_1
讀者有沒有發現,$a_0b_0,a_1b_0+a_0b_1$正好是
的前兩項.我們再讓$a_0$為0,$a_1$的下标減去1,則
\end{align*}變為
0,0,a_0b_0
我們發現,$a_0b_0$正好是
的第一個非零項.這就是一種遞推規律.