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Copy file name to clipboardExpand all lines: content/抽象代数/Galois.md
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@@ -11,7 +11,9 @@ title: Galois Theory
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- Let $E$ be a field. An **automorphism** of $E$ is a isomorphism of $E$ onto itself.
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- 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**.
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- 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 $F$.
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- 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,$\text{irr}(\alpha, F) = \text{irr}(\beta, F)$.
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- 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,$\text{irr}(\alpha, F) = \text{irr}(\beta, F)$.
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-**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 $E$, $F \leq E < K$, 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 $E$ is a **splitting field** for some finite set of polynomials.
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- Let $\sigma : F \rightarrow F$ be a field isomorphism; then $\sigma_x : F[x] \rightarrow F [x]$, defined by $\sigma_x(a_0 + a_1x + \cdots + a_n x^n ) = \sigma (a_0 ) + \sigma (a_1 )x + \cdots + \sigma (a_n )x^n$, is the **polynomial extension** of $\sigma$.
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## 3. 性质
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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 $F$.
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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 $\sigma (\alpha) = \beta$.
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7. Let $f(x) \in F[x]$. If $a, b \in F$ and $f(a + bi) = 0$, then $f(a - bi) = 0$.
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8. 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 $F$.
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9. if $\sigma : F \mapsto F$ is an isomorphism, then $\sigma_x : F[x] \mapsto F [x]$ is also an isomorphism.
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10. Let $K = F(\alpha)$, where $\alpha$ is algebraic over $F$, and let $\sigma : F \mapsto F'$ 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 $\varphi : F(\alpha) \mapsto F'(\beta)$ with $\sigma (a) = \varphi(a)$ for all $a \in F$ and $\varphi(\alpha) = \beta$.
11. Let $R$ and $R'$ be rings and let $N$ and $N'$ be ideals of $R$ and $R'$ respectively. Let $\phi$ be a homomorphism of $R$ into $R'$. Then $\phi$ induces a natural homomorphism $\phi^* : R/N \mapsto R'/N':$ $\phi^*(a+N)= \phi(a)+N'$ if $\phi[N] \subseteq N'$. $\phi[N] \subseteq N'$确保了同态映射$\phi^*$的良定义性。
9. Let $G$ and $G'$ be groups, and let $H$ and $H'$ be normal subgroups of $G$ and $G'$ respectively. Let $\phi$ be a homomorphism of $G$ into $G'$. Then $\phi$ induces a natural homomorphism $\phi^* : (G/H) \mapsto (G'/H'):$ $\phi^*(aH)= \phi(a)H'$ if $\phi[H] \subseteq H'$. $\phi[H] \subseteq H'$确保了同态映射$\phi^*$的良定义性。
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