00.3.体の様々な定義

体の同型写像

$E,E'$:体
$f:E \longrightarrow E' \quad$ (全単射)
$\forall x,y \in E$
$f(xy) = f(x)f(y)$
$f(x + y) = f(x) + f(y)$
このとき$f$を体の同型写像(Isomorphism)と呼ぶ.
$E = E'$ のとき$f$を自己同型写像(Automorphism)と呼び.
$Auto(E)$ と書く.

体の拡大

$E,F$:体
$E \supset F$
$\forall x,y \in F$
$x (+_E) y = x (+_F) y$
$x (*_E) y = x (*_F) y$
このとき
$F$を$E$の部分体
$E$を$F$の拡大体
と呼ぶ. $E,F,E',F'$:体
$E \supset F$
$E' \supset F'$
$\sigma_0:F \longrightarrow F'\quad$ (同型写像)
$\sigma:E \longrightarrow E'\quad$ (同型写像)
$\forall x \in F,\sigma_0(x) = \sigma(x)$ のとき
$\sigma$は $\sigma_0$ の延長
$\sigma_0$は $\sigma$ の制限
と呼ぶ.
$F = F',\sigma_0(x)=x$ のとき $\sigma$は$F$上の同型写像
さらに $E = E'$ のとき $\sigma$は$F$上の自己同型写像
$E$ と $E'$ は$F$上同型
と呼び.
$E \stackrel{\sigma}{\simeq} E'$ と書く. $E \stackrel{\sigma}{\simeq} E' \alpha,\beta \in E - F, \sigma(\alpha) = \beta$
この時 $\alpha$ と $\beta$ は "$F$ 上共役"という.

拡大次数

$E/F$
then $E$ is a vector space over $F$.
$E = \{\sum_{i=1}^n f_i e_i | e_i \in E, f_i \in F\}$
$E$ is a vector space over F.
$n$ is called 拡大次数.($n = \dim E = [E:F]$)
if $n \lt \infty$ then 有限次拡大
if $n = \infty$ then 無限次拡大

Example

$\mathbb{C}/\mathbb{R}$
$\mathbb{C} = \{a + bi|a, b \in \mathbb{R}\}$
$[\mathbb{C}:\mathbb{R}] = 2$

モニック(monic)な多項式

最高次数係数が"1"の多項式.

代数的な元

$F$:Field
$J_\alpha = \{f(x) \in F[x]|f(\alpha) = 0\}$
If
$J_\alpha \neq \{0\}$
Then $\alpha$ は $F$ 上、代数的(algebraic over $F$).
Else if
$J_\alpha = \{0\}$
Then $\alpha$ は $F$ 上、超越的(transcendental over $F$).
すなわち $\alpha$ が $F$ 上、代数的とは、 $F$ 係数多項式で $\alpha$ を解に持つものが存在するということ.

最小多項式(minimal polynomial)

$F$:Field
$\alpha$:Algebraic over $F$.
$J_\alpha = \{f(x) \in F[x]|f(\alpha) = 0\}$
then
$\exists p(x) \in F[x]$ such that $J_\alpha = (p(x))$ ($\because F[x]$ is PID)

$p(x)$ の最高次数係数を $a \in F$ とすると
$a^{-1}p(x)$ is called a minimal polynomial of $\alpha$ over $F$.
($\alpha$ の $F$ 上最小多項式).

体の元の次数

$F$:Field
$\alpha$: $F$ 上代数的
$p(x)$: $\alpha$ の $F$ 上、最小多項式.
then
$\text{deg} p(x)$ is called degree of $\alpha$ over $F$.
($\alpha$ の $F$ 上の次数.)

代数的拡大

$E/F$:Field
$\forall \alpha \in E,\alpha:$ is algebraic over $F$
then
$E$ is a algebraic extension of $F$

代数的閉体(Algebraically Closed Field)

$F$:Field
$\forall f(x) \in F[x], f(\alpha) = 0 \Rightarrow \alpha \in F$
($\Leftrightarrow \forall f(x) \in F[x], \exists a, \alpha_1, \cdots , \alpha_n \in F, \text{ s.t. } f(x) = a(x - \alpha_1)\cdots(x - \alpha_n)$)
Then
$E$ is a Algebraically Closed Field.

代数的閉包(Algebraic Closure)

$E/F$:Algebraic Extension
$E$:Algebraic Closed Field
Then
$E$ is a Algebraic Closure over $F$

分解体(Splitting Field)・最小分解体(Minimal Splitteing Field)

$E/F$:Field
$f(x) \in F[x], \text{deg} f(x) = n \gt 0$
$f(\alpha) = 0 \Rightarrow \alpha \in E$
then $E$ is called a Splitting Field of $f(x)$ over $F$.
$E = F(\alpha_1, \cdots , \alpha_n)$
then $E$ is called a Minimal Splitting Field of $f(x)$.
$E = \bar F$

重根(Multiple Root)

$F$:Field
$f(x) \in \bar F[x]$
$\alpha \in \bar F,f(\alpha) = 0$
if $(x - \alpha)^2 | f(x)$
then
$\alpha$ is a 重根 of $f(x)$

導多項式

$F$:Field
$f(x) \in F[x]$
$f(x) = a_n x^n + \cdots + a_1 x + a_0$
Then $f'(x) = n a_n x^{n-1} + \cdots + a_1$ is called a derivative of $f(x)$

分離多項式(Separable Polynomial)

$F$:Field
$f(x) \in F[x]$
if $\alpha \in \bar F,f(\alpha) = 0 \Rightarrow (x - \alpha)^2 \nmid f(x)$
then $f(x)$ is a Separable Polynomial.(=分離多項式)
if $\alpha \in \bar F,f(\alpha) = 0 \Rightarrow (x - \alpha)^2 \mid f(x)$
then $f(x)$ is a Inseparable Polynomial.(=非分離多項式)

分離的元(Separable)

$E,F$:Field.
$E/F$:Algebraic Extension.
$\alpha \in E$:Algebraic over $F$.
$f(x) \in F[x]$:Minimal Polynomial of $\alpha$.
if $f(x)$:Separable Polynomial, then
$\alpha$ is Separable over $F$.

分離的拡大(Separable Extension)

$E,F$:Field.
$E/F$:Algebraic Extension.
if $\forall \alpha \in E$ is Separable over $F$, then
$E$ is a Separable Extension of $F$.

標数(Characteristic)

$F$:Field
$1_F \in F$:identity element of $F$
if $\exists n \in \mathbb{Z}, n \cdot 1_F = 0$ then
the smallest number of $n$ is called characteristic of $F$
if $\nexists n \in \mathbb{Z}, n \cdot 1_F = 0$ then
characteristic of $F$ is defined by 0

完全体(Perfect)

$F$:Field
then
$F$:完全体
is defined by
$\forall f(x) \in F[x]$:Irreducible $\Rightarrow f(x)$:Separable
$\Leftrightarrow$
$\forall E/F$:Algebraic Extension $\Rightarrow E/F$:Separable Extension

正規拡大

$E,F$:Field
$E/F$:代数的拡大
$\forall \alpha_1 \in E,\exists f(x) \in F[x] \text{ s.t. }f(x)$:既約
$\text{dim}f(x) = n$
then
$\exists \alpha_2, \cdots , \alpha_n \in E,f(x) = a(x - \alpha_1)(x - \alpha_2) \cdots (x - \alpha_n)$

$\forall \alpha \in E \Rightarrow \forall \alpha'(:\text{ 共役 of } \alpha) \in E$

$E/F$ is called 正規拡大
.

ガロア拡大

$E,F$:Field
$E/F$:分離的,正規拡大
then
$E/F$ is called ガロア拡大

ガロア群

$E/F$:ガロア拡大
$E$の$F$上の自己同型写像全体をなす群をガロア群と呼び
$Gal(E/F)$
と書く

固定体(Fixed Field)

$F$:Field
$G \subset Auto(F)$
then, define
$F^G = \{x \in F| \forall g \in G, g(x) = x\}$
$F^G$ is called Fixed Field(固定体).

アーベル拡大

$Gal(E/F)$ がアーベル群のとき、アーベル拡大と呼ぶ

可解拡大(solvable extension)

$E/F$:ガロア拡大
then
$E/F$ is defined as a solvable extension, if $Gal(E/F)$ is a solvable Group.