抽象代数入门(三)

一些域上多项式的概念

prime(素)

Dfn: p\in\mathbb{F}[z], p/(p_1p_2)\Rightarrow p/p_1\ or\ p/p_2. Then p is prime.

 

Unit(单位)

Dfn: If p\in\mathbb{F}[z] is unit, and p\not=0, then: \forall g\in\mathbb{F}[z], s.t. (g)=(pg).  A given field may have infinite units.(g) 表示minimal and monoic polynomial of g.

 

Relative prime(相对素)

Dfn:p_1, p_2 is relative prime if p_1, p_2\in \mathbb{F}[z], gcd(p1, p_2)=1. gcd 是 great common divisor(最大公约数)的简写。

 

Irreducible(不可约) 

p\in\mathbb{F}[z], deg(p)\geq1, p=p_1p_2\Rightarrow p_1\ is\ unit\ or\ p_2\ is\ unit.

Theorem:

If p\in \mathbb{F}[z], p: \ irreducible \leftrightharpoons p:\ prime.

Proof: irreducible\Rightarrowprime. If p is irreducible, then p/(p_1p_2)\Rightarrow p/p_1\ or\ p/p_2. if p is not divisor of p_1, then we will show p/p_2.

略哈,现在不想写。

 

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}

p=c_0 + c_1x+...+c_sx^s.

p(A)=c_0 + c_1A+...+c_sA^s.

Annihilation

Dfn: p "annihilates" \mathbb{F}^n means p\in\mathbb{F}[x], A\in\mathbb{R}^{n\times n}, if\ p(A)=0\Rightarrow p\ annihilates \mathbb{F}^n.

i.e. T:=ann(\mathbb{F}^n)=\left\{q\in\mathbb{F}[x]|qv=0,\forall v\in \mathbb{F}^n \right\}, T is ideal of \mathbb{F}[x]

proof: 

  1. p_1,p_2\in T, (p_1+p_2)v=p_1v+p_2v=0, \forall v\in\mathbb{F}^n\Rightarrow (p_1+p_2)\in T
  2. p\in T,q\in \mathbb{F}^n, pv=0,\forall v\in\mathbb{F}^n. qpv=q(pv)=0\Rightarrow qp\in T.

 

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