Galois Theory

1. 前置

• 域论(有限域)

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 $\sigma :E\to K$ 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 $F\le K$ 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,$\text{irr}(\alpha ,F)=\text{irr}(\beta ,F)$.
• 分裂域：Let $F$ be a field and $P=\{f_1(x),f_2(x),\cdots ,f_s(x)\}$ be a finite set of polynomials in $F[x]$. An extension field $K$ of $F$ is a splitting field of $P$ over $F$ if every polynomial $f_k(x)\in P$ factors into linear factors in $K[x]$ and for any intermediate field $E$, $F\le E<K$, at least one polynomial $f_j(x)\in P$ does not factor into linear factors in $E[x]$. A field $K$ is a splitting field for $F$ if $K$ is a splitting field for some finite set of polynomials. （本节只讨论有限分裂域（finite splitting field），即$P$是有限集）
• Let $\sigma :F\to F$ be a field isomorphism; then ${\sigma }_x:F[x]\to F[x]$, defined by ${\sigma }_x(a_0+a_{1}x+\cdots +a_n{x}^n)=\sigma (a_0)+\sigma (a_1)x+\cdots +\sigma (a_n)x^n$, is the polynomial extension of $\sigma$.
• Let $E$ be an extension field of $F$. A polynomial $f(x)\in F[x]$ splits in $E$ if it factors into linear factors in $E[x]$.
• Let $f(x)\in F[x]$, and let $\alpha$ be a zero of $f(x)$ in a splitting field $E$ over $F$. If $\nu$ is the largest positive integer such that $(x-\alpha {)}^{\nu }$ is a factor of $f(x)$ in $E[x]$, then $\alpha$ is a zero of $f(x)$ with multiplicity $\nu$. $\nu$与分裂域的选择无关，在分裂域的扩域上也是同样的重数。
• An irreducible polynomial $f(x)\in F[x]$ of degree $n$ is separable if in the splitting field $K$ of $f(x)$ over $F$, $f(x)$ has $n$ distinct zeros. An element $\alpha$ in an extension field of $F$ is separable if $\text{irr}(\alpha ,F)$ is a separable polynomial. A field extension $F\le E$ is separable if every $\alpha \in E$ is separable over $F$. If every finite extension of a field $F$ is separable, then $F$ is perfect. （可分拓展是逐元素地考虑不可约多项式是否在分裂域中无重根）
• A finite extension $E$ of $F$ is a normal extension of $F$ if $E$ is a separable splitting field over $F$. If $E$ is a normal extension of $F$, then $G(E/F)$ is the Galois group of $E$ over $F$. The Galois group is sometimes denoted by $\text{Gal}(E/F)$.
• Let $K$ be a normal extension of $F$, $E$ an intermediate field of the extension, and $H$ a subgroup of $G(K/F)$. The set of all $\alpha \in K$ such that each element of $H$ fixes $\alpha$ is an intermediate field of the extension $K$ over $F$, and it is called the fixed field for $H$. We write $K_H$ to denote the fixed field for $H$. We let $\lambda (E)$ be the set of all $\sigma \in G(K/F)$ that fix all the elements of $E$, that is, $\lambda (E)=G(K/E)$. We call $\lambda (E)$ the group of $E$. If $K$ is the splitting field of $f(x)\in F[x]$, then we say that $G(K/F)$ is the group of the polynomial $f(x)$.

一个域上的多项式的重数只依赖于多项式本身，
因为裂域在同构的意义下是唯一的，
而分裂域的扩域不影响线性因子分解。

3. 性质

1. (域上的所有自同构映射构成一个群)Let $E$ be a field. Then the set of all automorphisms of $E$ is a group under composition.
2. (域上自同构的不动点构成一个子域)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. (域上若干自同构的公共不动点构成一个子域)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. (域的自同构群里的所有保持子域不变的自同构构成一个子群) 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 $E_{G(E/F)}$ （不动域）.
5. (共轭同构定理，The Conjugation Isomorphism) Let $F$ be a field, $K$ an extension field of $F$, and $\alpha ,\beta \in K$ algebraic over $F$ with $deg(\alpha ,F)=n$. The map ${\psi }_{\alpha ,\beta }:F(\alpha )\to F(\beta )$ 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 $c_i\in F$, 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 $f(x)\in \mathbb{R}[x]$. If $a,b\in F$ and $f(a+bi)=0$, then $f(a-bi)=0$.
8. (分裂域存在)Let $F$ be a field and $P=\{f_1(x),f_2(x),\cdots ,f_s(x)\}$ a finite set of polynomials in $F[x]$. Then there is a splitting field $K$ of $P$ over $F$. Furthermore $K$ is a finite extension of $F$.
9. if $\sigma :F\mapsto F$ is an isomorphism, then ${\sigma }_x:F[x]\mapsto F[x]$ is also an isomorphism.
10. Let $K=F(\alpha )$, where $\alpha$ is algebraic over $F$, and let $\sigma :F\mapsto F^{\prime }$ be a field isomorphism. If $K$ is an extension field of $F$ and $\beta \in K$ is a zero of ${\sigma }_x(\text{irr}(\alpha ,F))$, then there is a unique isomorphism $\phi :F(\alpha )\mapsto F^{\prime }(\beta )$ with $\sigma (a)=\phi (a)$ for all $a\in F$ and $\phi (\alpha )=\beta$.
11. (Isomorphism Extension Theorem) Let $K=F({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n)$ be a finite extension field of $F$, and let $\sigma :F\mapsto F$ be a field isomorphism. If $K^{\prime }$ contains a splitting field of $P=\{{\sigma }_x(\text{irr}({\alpha }_k,F))|1\le k\le n\}$ over $F$ , then $\sigma$ can be extended to an isomorphism $\tau$ mapping $K$ onto a subfield of $K^{\prime }$
12. (分裂域在同构意义下唯一) Let $F$ be a field, $P=\{f_1,f_2,\cdots ,f_s\}\subseteq F[x]$ a finite set of polynomials, and both $K$ and $K^{\prime }$ splitting fields of $P$ over $F$. Then there is an isomorphism $\sigma :K\to K^{\prime }$ , which is the identity map on $F$.
13. Let $E$ be a finite extension of the field $F$. Then $E$ is the splitting field of some finite set of polynomials in $F[x]$ if and only if for every field extension $K$ over $E$ and for every isomorphism $\sigma$ that fixes all the elements of $F$ and maps $E$ onto a subfield of $K$, $\sigma$ is an automorphism of $E$.
14. If $K$ is a finite splitting field over $F$ and $K$ contains one zero of an irreducible polynomial $f(x)\in F[x]$, then $f(x)$ splits in $K[x]$.
15. Let $F\le E\le K$ be fields with $K$ a finite splitting field over $F$. Then $E$ is a splitting field over $F$ if and only if every isomorphism $\sigma$ that fixes $F$ and maps $E$ to a subfield of $K$ is an automorphism of $E$.
16. Let $f(x)$ be an irreducible polynomial of degree $n$ with coefficients in a field $F$ of characteristic zero. Then $f(x)$ contains $n$ distinct zeros in the splitting field for $f(x)$ over $F$.
17. Let $F$ be a finite field of characteristic $p$. Any irreducible polynomial $f(x)\in F[x]$ has $k=\text{deg}(f(x))$ distinct zeros in its splitting field.
18. Every field of characteristic $0$ is perfect and every finite field is perfect.
19. Let $K$ be a separable extension of the field $F$ and $E$ an intermediate field. Then both the extensions $K$ over $E$ and $E$ over $F$ are separable.
20. Let $K$ be a splitting field over $F$ and $F\le E\le K$. If $K$ is a separable extension over $F$, then the number of isomorphisms that map $E$ onto a subfield of $K$ that fix all the elements of $F$ is $[E:F]$.
21. Let $E$ be a separable splitting field over $F$. Then $|G(E/F)|=[E:F]$.
22. Let $E$ be a splitting field over $F$ where $F$ is either a field of characteristic 0 or a finite field. Then $|G(E/F)|=[E:F]$.
23. (Primitive Element Theorem) Let $E$ be a finite separable extension of a field $F$. Then there is an $\alpha \in E$ such that $E=F(\alpha )$. Any such element $\alpha$ is called a primitive element.
24. If $F$ is either a finite field or a field of characteristic 0, then every finite extension of $F$ is a simple extension.
25. Let $K$ be a normal extension of $F$ and let $E$ be an intermediate field of the extension, $F\le E\le K$. Then $K$ is a normal extension of $E$ and $|\text{Gal}(K/E)|=[K:E]$.
26. If $F\le E\le K$ where $K$ is a normal extension of $F$, then $G(K/E)$ is a subgroup of $G(K/F)$ with index $(\text{Gal}(K/F):\text{Gal}(K/E))=[E:F]$.
27. Let $K$ be a normal extension of a field $F$ and $E$ an intermediate field. The fixed field for the set of all automorphisms of $K$ that fix $E$ is exactly $E$. That is, $E=K_{\lambda (E)}$.
28. Let $K$ be a normal extension of a field $F$ and $E$ an intermediate field. The degree of the extension $K$ over $E$ is the order of the group $\lambda (E)$:
    $[K:E]=|\lambda (E)|=|G(K/E)|$.
    Furthermore, the number of left cosets of $\lambda (E)$ in $G(K/F)$ is the degree of the extension of $E$ over $F$. That is, $(G(K/F):\lambda (E))=[E:F]$.
29. Let $K$ be a normal extension of a field $F$ and $H$ a subgroup of the Galois group $G(K/F)$. The subgroup of $G(K/F)$ that fixes all the elements fixed by $K_H$ is exactly $H$. That is, $\lambda (K_H)=H$.
30. Let $K$ be a normal extension of a field $F$ and $E$ an intermediate field of the extension. Then $E$ is a normal extension of $F$ if and only if $\lambda (E)$ is a normal subgroup of $G(K/F)$. Furthermore, if $E$ is a normal extension of $F$, then $G(E/F)$ is isomorphic with $G(K/F)/G(K/E)$.
31. Let $K$ be a normal extension of $F$, with $E_1$ and $E_2$ intermediate fields. Then $E_1$ is a subfield of $E_2$ if and only if $\lambda (E_2)$ is a subgroup of $\lambda (E_1)$.

4. 关系图

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。

下图对任意的一个不可约多项式都成立。特别的对于$f(x)$的不可约因子也成立

graph LR
    F["`$$F$$`"] -->|构造域上多项式环| Fx["`$$F[x]$$`"] --> E["`$$E$$`"]
    Fx -->|因子环（域）| Fx_p["`$$F[x]/\langle p(x) \rangle$$`"]
    Fx_p -->|子域| Fx_p_F["`$$\{a+\langle p(x) \rangle|a \in F\}$$`"]
    F <------> |同构|Fx_p_F
    f_x["`$$f(x)$$`"] --> |不可约因子| p_x["`$$p(x)$$`"] --> Fx_p
    z_p["`$$x+\langle p(x)\rangle $$`"] -->|零点| Fx_p

4.2. 单扩域(单拓域)

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。$\varphi (x):F[x]\mapsto E$为估值环同态映射，$F[\alpha ]$为$\varphi$的像（即$F$的单扩域）

graph LR
    F["`$$F$$`"] -->|构造域上多项式环| Fx["`$$F[x]$$`"] -->|估值同态映射| E["`$$E$$`"]
    Fx -->|因子环（域）| Fx_p["`$$F[x]/\langle p(x) \rangle$$`"]
    Fx_p -->|子域| Fx_p_F["`$$\{a+\langle p(x) \rangle|a \in F\}$$`"]
    F <------> |同构|Fx_p_F
    z_p["`$$x+\langle p(x)\rangle $$`"] -->|零点| Fx_p
    E_sub["`$$F[\alpha]$$`"] -->|子域| E
    Fx_p <--> |同构| E_sub