Quartz sync: Nov 4, 2025, 7:58 PM

Author: zeroDtree

content/抽象代数/Galois.md

@@ -2,28 +2,30 @@
 title: Galois Theory
 ---
 
-## 前置
+## 1. 前置
 
 - [[抽象代数/域论(域拓展)]]
 
-## 定义
+## 2. 定义
 
 - Let $E$ be a field. An **automorphism** of $E$ is a isomorphism of $E$ onto itself.
 - If $E$ and $K$ are both field extensions of a field $F$ and $\sigma : E \rightarrow K$ is a field isomorphism, then an element $\alpha \in E$ is **fixed** by $\sigma$ if $\sigma (\alpha) = \alpha$. An element $\alpha \in E$ is **fixed** by a collection of isomorphisms if $\alpha$ is fixed by every isomorphism in the collection. A subset $L$ of $E$ is **fixed** by a collection of isomorphisms if every $\alpha \in L$ is fixed by the collection. Often write **remains fixed** instead of simply **fixed**.
 - Let $F \leq K$ be a field extension. The set $G(K/F)$ is the set of all automorphisms of the field $K$ that fix every element of the field $F$.
 - Let $E$ be an algebraic extension of the field $F$. Two elements $\alpha$ and $\beta$ in $E$ are conjugates over $F$, if both have the same minimal polynomial over $F$. That is,$\text{irr}(\alpha, F) = \text{irr}(\beta, F)$.
 
-## 性质
+## 3. 性质
 
 1. Let $E$ be a field. Then the set of all automorphisms of $E$ is a group under composition.
 2. Let $\sigma$ be an automorphism of the field $E$. Then the set $E_\sigma$ of all the elements $a \in E$ that remain fixed by $\sigma$ forms a subfield of $E$.
 3. Let $\{ \sigma_i | i \in I \}$ be a collection of automorphisms of a field $E$. Then the set $E_{\{\sigma_i\}}$ , of all $a \in E$ that remain fixed by every $\sigma_i$, for $i \in I$, is a subfield of $E$. 因为域的任意交集还是域。
 4. Let $E$ be a field and let $F$ be a subfield of $E$. Then the set $G(E/F)$ of all automorphisms that fix all the elements of $F$ is a subgroup of the automorphism group of $E$. Furthermore, $F$ is a subfield of $E_{G(E/F)}$.
 5. **(The Conjugation Isomorphism)** Let $F$ be a field, $K$ an extension field of $F$, and $\alpha, \beta ∈ K$ algebraic over $F$ with $deg(\alpha, F) = n$. The map $\psi_{\alpha,\beta} : F(\alpha) → F(\beta)$ defined by $\psi_{\alpha,\beta} (c_0 + c_1 \alpha + c_2 \alpha^2 + \cdots + c_{n-1} \alpha^{n-1} ) = c_0 + c_1 \beta + c_2 \beta^2 + \cdots + c_{n-1} \beta^{n-1} $, for $c_i ∈ F$, is an isomorphism of $F(\alpha)$ onto $F(\beta)$ if and only if $\alpha$ and $\beta$ are conjugate over $F$.
+6. Let $K$ be a field extension of $F$ with $\alpha \in K$ algebraic over $F$. Suppose that $\psi$ is an isomorphism of $F(\alpha)$ onto a subfield of $K$, with the property that every element of $F$ is fixed by $\psi$. Then $\psi$ maps $\alpha$ to a conjugate over $F$ of $\alpha$. Conversely, if $\beta \in K$ is conjugate over $F$ with $\alpha$, then there is a unique isomorphism $\psi_{\alpha,\beta}$ mapping $F(\alpha)$ onto a subfield of $K$ with the properties that each $a \in F$ is fixed by $\sigma$ and $\sigma (\alpha) = \beta$.
+7. Let $f(x) \in F[x]$. If $a, b \in F$ and $f(a + bi) = 0$, then $f(a - bi) = 0$.
 
-## 关系图
+## 4. 关系图
 
-### Kronecker 定理
+### 4.1. Kronecker 定理
 
 设$F$为一个域，$E$为$F$的扩域，$f(x)$为$F[x]$上的一个非常量多项式，$\alpha \in E$为$F$的代数元, $p(x)$为$f(x)$的不可约因子。
 
@@ -39,7 +41,7 @@ graph LR
     z_p["`$$x+\langle p(x)\rangle $$`"] -->|零点| Fx_p
 ```
 
-### 单扩域
+### 4.2. 单扩域
 
 设$F$为一个域，$E$为$F$的扩域，$\alpha \in E$为$F$的代数元, $p(x):=\text{irr}(\alpha, F)$为$\alpha$的不可约多项式。$\phi(x):F[x]\mapsto E$为估值环同态映射，$F[\alpha]$为$\phi$的像（即$F$的单扩域）