Field(域)

Dfn: ”Field“ is a commutative ring $(\mathbf{R}, +, .)$ , s.t. every non-zero element has inverse.

這個交換環對於乘法和加法都滿足交換性，而且對於所有非零元素都存在逆。

對加法逆，即 $(\mathbf{R}, +), \forall v\in \mathbf{R}, v\not=0, \exists u\in\mathbf{R}, s.t. \quad v + u=0;$
對乘法逆，即 $(\mathbf{R}, *),\forall v\in \mathbf{R}, v\not=0, \exists u\in\mathbf{R}, s.t. \quad v*u=1.$

這兩個都滿足，u是對應元素的逆。加法的逆寫成-v, 乘法的逆寫成 $v^{-1}$ 都是可以的，都是符號表示。

Polynomials over a field

Dfn: Let $(\mathbb{F}, +, *)$ be a field. A polynomial over a field $\mathbb{F}$ is an equation of the form :, where the coefficients $a_0, a_1, ..., a_m\in \mathbb{F}$ . The set of all polynomials over a field $\mathbb{F}$ is denoted $\mathbb{F}[x]$ .

ideal(理想)

Dfn: "Idela" of $\mathbb{F}[z]$ is a subset $\mathbf{T}$ of $\mathbb{F}[z]$ s.t. it is closed under addition and closed under multiplication by any element of $\mathbb{F}[z]$ .

closed under addition: $\forall u_1, u_2\in \mathbf{T}, s.t. \quad u_1+u_2\in\mathbf{T}$
closed under multiplication: $\exists r\in \mathbb{F}[z], \forall u\in\mathbf{T}, s.t. \quad r*u\in\mathbf{T}$

i.e. the set of even integers is an ideal in the ring of integers $\mathbf{Z}$ . Given an ideal $\mathbf{T}$ , it is possible to define a quotient ring $\mathbb{F}[z]/\mathbf{T}$ .

i.e. $\mathbf{T}=\left\{p\in\mathbb{F}[z]|p=(z^2+1)q, \forall q\in\mathbb{F}[z] \right\}.$ is ideal.

$\left.\begin{matrix} p_1=(z^2+1)q_1 \\ p_2=(z^2+1)q_2 \end{matrix}\right\},\quad p_1+p_2=(z^2+1)(q_1+q_2)\in T,\ where\ q_1,q_2\in\mathbb{F}[z]$
$p\in\mathbf{T},\ h\in\mathbb{F}[z],then\ p=(z^2+1)q, where \ q\in\mathbb{F}[z], hp=(z^2+1)hq\in\mathbf{T}$ . proved

Module(模)

Dfn: a module is one of the fundamental algebraic structures used in abstract algebra. A module over a ring is a generalization of the notion of vector space over a field.

環上的一個模，域上的一個向量空間。在模中的係數是來自於環的，而向量空間中的係數是來自於域的。（域的定義更爲嚴格，比之於環），係數的尋去空間更爲通用，因此說模是向量空間的泛化。

Integral Domain(整域)

Dfn: 也叫整環，指的是不含零因子(divisor)的交換環。環中乘法的單位元通常和加法的單位元不同，整環是整數環的抽象化。關於交換環(commutative ring)的定義可以參看抽象代數入門（一）。

multiplicative identity element: $a*e=e*a=1, \forall a\in\mathbf{R}, e\ is\ the\ identity\ element$ .
No divisors of 0, $\forall (a, b) \in\mathbf{R}, s.t. a*b=0\Rightarrow (a=0\cup b=0)$ or $\forall (a, b, c) \in\mathbf{R}, if\ a*c=b*c\cup c\not= 0\Rightarrow (a=b)$

i.e. $(\mathbb{Z}, +, *)$ is an intergral domain.

Principal Ideal Ring(主理想環)

Dfn: 每個理想都可以由單個元素生成的環，寫成left ideal xR or right ideal Rx 的形式，其中 x 是環 R 中的一個元素。

Dfn: 如果交換的理想環R 同時也是整環，那麼稱之爲主理想域(principal ideal domain), 簡寫成 P.I.D.

i.e. 整數環是主理想域，Eucildean Ring 一般都是。

Polynomial Ring(多項式環)

多項式環是對初等數學中多項式的泛化，通常表示成 $\mathbf{R}[X]$ 。

Dfn: The polynomial ring, $\mathbf{R}[X]$ , in over $\mathbf{R}$ is defined as:

, is the coefficient of , 只是一個符號，稱爲 powers of .

The ideal generated by a finite set of polynomials $\left\{p_1, p_2,..., p_m \right\}\in\mathbb{F}[x]$ defined as:

$\left$p_1, p_2, ..., p_m\right$:=\left\{q_1p_1+q_2p_2+...+q_mp_m|\forall p_i\in\mathbb{F}[x] \right\}$

Monoic Polynomial(單多項式)

Dfn: a single-variable polynomial(univariate polynomial) in which the leading coefficient is equal to 1. Which can be written in the form:

$p(x)=x_n+a_{n-1}x_{n-1} + a_{n-2}x_{n-2}+...+a_1x_1+a_0$

Unique Factorization Domain(唯一分解環)

Dfn: 唯一分解環是一個整環(integral domain), 其元素都可以表示成有限個不可約元素乘積的形式，並在允許重排下唯一，相當於滿足基本算術定理的整環。簡寫爲UFD.

A integral domain R is said to be a unique factorization domain if it has the following factorization properties:

Every nonzero nonunit element $r\in\mathbf{R}$ can be written as a product of a finite number of irreducible elements $r\in p_1p_2...p_s$ .
The factorization into irreducible elements is unique in the sense that if and are two factorizations. Then s=t and after a suitable reindexing of the factor, .