一些域上多項式的概念
prime(素)
Dfn: ![p\in\mathbb{F}[z], p/(p_1p_2)\Rightarrow p/p_1\ or\ p/p_2. "/" \ means\ exact \ divison.](https://pic1.xuehuaimg.com/proxy/csdn/https://private.codecogs.com/gif.latex?p%5Cin%5Cmathbb%7BF%7D%5Bz%5D%2C%20p/%28p_1p_2%29%5CRightarrow%20p/p_1%5C%20or%5C%20p/p_2.%20%22/%22%20%5C%20means%5C%20exact%20%5C%20divison.)
Unit(單位)
Dfn： If ![p\in\mathbb{F}[z]](https://pic1.xuehuaimg.com/proxy/csdn/https://private.codecogs.com/gif.latex?p%5Cin%5Cmathbb%7BF%7D%5Bz%5D)
![\forall g\in\mathbb{F}[z], s.t. (g)=(pg)](https://pic1.xuehuaimg.com/proxy/csdn/https://private.codecogs.com/gif.latex?%5Cforall%20g%5Cin%5Cmathbb%7BF%7D%5Bz%5D%2C%20s.t.%20%28g%29%3D%28pg%29)
Relative prime(相對素)
Dfn:
![p_1, p_2\in \mathbb{F}[z], gcd(p1, p_2)=1](https://pic1.xuehuaimg.com/proxy/csdn/https://private.codecogs.com/gif.latex?p_1%2C%20p_2%5Cin%20%5Cmathbb%7BF%7D%5Bz%5D%2C%20gcd%28p1%2C%20p_2%29%3D1)
Irreducible(不可約)
![p\in\mathbb{F}[z], deg(p)\geq1, p=p_1p_2\Rightarrow p_1\ is\ unit\ or\ p_2\ is\ unit.](https://pic1.xuehuaimg.com/proxy/csdn/https://private.codecogs.com/gif.latex?p%5Cin%5Cmathbb%7BF%7D%5Bz%5D%2C%20deg%28p%29%5Cgeq1%2C%20p%3Dp_1p_2%5CRightarrow%20p_1%5C%20is%5C%20unit%5C%20or%5C%20p_2%5C%20is%5C%20unit.)
Theorem:
If
![p\in \mathbb{F}[z], p: \ irreducible \leftrightharpoons p:\ prime.](https://pic1.xuehuaimg.com/proxy/csdn/https://private.codecogs.com/gif.latex?p%5Cin%20%5Cmathbb%7BF%7D%5Bz%5D%2C%20p%3A%20%5C%20irreducible%20%5Cleftrightharpoons%20p%3A%5C%20prime.)
Proof: irreducible
略哈，現在不想寫。
Definition:
![A\in\mathbb{R}^{n\times n}, \mathbb{R}[x]\times\mathbb{R}^n\rightarrow \mathbb{R}^n. pv = p(A)v. \mathbb{F}=\mathbb{R}, \mathbb{C}](https://pic1.xuehuaimg.com/proxy/csdn/https://private.codecogs.com/gif.latex?A%5Cin%5Cmathbb%7BR%7D%5E%7Bn%5Ctimes%20n%7D%2C%20%5Cmathbb%7BR%7D%5Bx%5D%5Ctimes%5Cmathbb%7BR%7D%5En%5Crightarrow%20%5Cmathbb%7BR%7D%5En.%20pv%20%3D%20p%28A%29v.%20%5Cmathbb%7BF%7D%3D%5Cmathbb%7BR%7D%2C%20%5Cmathbb%7BC%7D)
Annihilation
Dfn: p "annihilates" 
![p\in\mathbb{F}[x], A\in\mathbb{R}^{n\times n}, if\ p(A)=0\Rightarrow p\ annihilates \mathbb{F}^n](https://pic1.xuehuaimg.com/proxy/csdn/https://private.codecogs.com/gif.latex?p%5Cin%5Cmathbb%7BF%7D%5Bx%5D%2C%20A%5Cin%5Cmathbb%7BR%7D%5E%7Bn%5Ctimes%20n%7D%2C%20if%5C%20p%28A%29%3D0%5CRightarrow%20p%5C%20annihilates%20%5Cmathbb%7BF%7D%5En)
i.e. ![T:=ann(\mathbb{F}^n)=\left\{q\in\mathbb{F}[x]|qv=0,\forall v\in \mathbb{F}^n \right\}](https://pic1.xuehuaimg.com/proxy/csdn/https://private.codecogs.com/gif.latex?T%3A%3Dann%28%5Cmathbb%7BF%7D%5En%29%3D%5Cleft%5C%7Bq%5Cin%5Cmathbb%7BF%7D%5Bx%5D%7Cqv%3D0%2C%5Cforall%20v%5Cin%20%5Cmathbb%7BF%7D%5En%20%5Cright%5C%7D)
![\mathbb{F}[x]](https://pic1.xuehuaimg.com/proxy/csdn/https://private.codecogs.com/gif.latex?%5Cmathbb%7BF%7D%5Bx%5D)
proof:
.
.
抽象代數入門（三）
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