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Recall from Galois Extensions, we listed two conditions that a field extension must be satisfied, in order for it to be called a Galois extension. Here we dive into the details of the two conditions.
Let \(E/F\) be a field extension. \(E/F\) is called a normal extension if \(\text{Aut}(E/F) = \text{Emb}(E/F)\).
Let \(E/F\) be a finite field extension and \(K\) be an intermediate field between \(E\) and \(F\). Given \(\rho \in \text{Emb}(K/F)\) and \(\tilde \rho\in \text{Emb}(E/F)\), \(\tilde\rho\) is called an extension of \(\rho\) to \(E\) if \(\tilde\rho\big\vert_K = \rho\).
這個定義，對於任何兩個值域有包含關係的函數都可以用！
當 \(\tilde{\rho}\) 的輸入是 \(k\in K\)，他的行為和 \(\rho\) 一模一樣！
Let \(E/F\) be a finite field extension and let \(K\) be an intermediate field bewteen \(E\) and \(F\). For all \(\rho\in\text{Emb}(K/F)\), there exists an extension of \(\rho\) to \(E\). Moreover, if \(F\) is of characteristic zero, then there are \([E:K]\) such extensions.
Proof
Let \(E=K(\alpha_1, \cdots, \alpha_k)\). We will show that \(\rho\) can be extended to \(K(\alpha_1)\); repeating this process, \(\rho\) can be extended to \(E\).
Let \(g(x) = \text{Irr}(\alpha_1, K)(x)\), then \(g^\rho(x)\) is also irreducible in \(K^{\rho}[x]\). Let \(\beta\) be a zero of \(g^\rho(x)\) in \(\bar F\). Then
\[\begin{gather} \tilde\rho: &K(\alpha_1)& \to &K[x]/\langle g(x)\rangle& \to &K^\rho[x]/\langle g^{\rho}(x)\rangle& \to &K^\rho(\beta)& \\ &f(\alpha_1)& \to &f(x)& \to &f^\rho(x)& \to &f^\rho(\beta)& \end{gather}\]gives an \(F\)-embedding of \(K(\alpha_1)\) so that \(\tilde{\rho}\big\vert_K = \rho\), and \(\tilde{\rho}(\alpha_1) = \beta\).
When \(F\) is of characteristic zero, we may assume that \(E=K(\alpha_1)\) by the primitive element theorem. In this case, \(g^{\rho}(x)\) contains \([E:K]\) distinct zeros in \(\bar F\) and each zero determiines a unique extension of \(\rho\). ◼
每個 extension 打到不同的 zero！
原來 char 為零是為了 separable！
補 \(g^{\rho}(x)\) 的定義…
If a field extension of \(F\), \(E\), is obtained by adjoining all zeros of a polynomial \(f(x)\in F[x]\) in \(\bar F\), then \(E\) is called the splitting field of \(f(x)\) over \(F\), denoted by \(F_f\). In other words, \(F_f\) is the smallest field extension of \(F\) in \(\bar F\) in which \(f(x)\) splits.
split 代表 \(f(x)\) 可以寫成 linear factor 的乘積。
\(f(x)\) 未必要 irreducible。
Let \(E\) be a finite field extension over \(F\). The following are equivalent:
- \(E/F\) is a normal extension.
- \(E\) is the splitting field of some polynomial over \(F\).
- For all elements in \(E\), its irreducible polynomial over \(F\) splits over \(E\).
\(F\) needs not be of characteristic zero.
For all elements in \(E\), its irreducible polynomial over \(F\) splits over \(E\). 說明了 \(E\) 是所有 \(\text{Irr}(\alpha, F)(x), \alpha \in E\) 的 splitting field！
然後呀，通常說 splitting field 的時候，會講明 \(E\) 是哪個或哪些多項式的 splitting field；說 normal extension 時，是哪些多項式並未表明！（像前一段那句）
Let \(E/F\) be a finite field extension and \(\text{char}F = 0\). Then \(E/F\) is a Galois extension iff \(E\) is the splitting field of some polynomial over \(F\).
因為當 \(char F=0\)，\(E/F\) 是 Galois extension iff \(E/F\) 是 normal！
The smallest field extension \(E'/F\) such that \(E/F \le E'/F\) and \(E'/F\) is a normal extension, called the normal closure of \(E/F\).
若 \(E=F(\alpha)\)，\(E/F\) 的 normal cloure 正是 \(\text{Irr}(\alpha, F)(x)\) over \(F\) 的 splitting field！
A finite extension \(E/F\) is called a separable extension if \(\vert \text{Emb}(E/F)\vert = [E:F]\).
Let \(E/F\) be a finite field extension. The following are eqivalent:
- \(E/F\) is a separable extension.
- \(E\) is obtained from \(F\) by joining some zeros of a separable polynomial in \(\bar F\).
- For all elements in \(E\), its irreducible polynomial over \(F\) is separable.
For a finite extension \(E/F\), the following are equivalent:
- \(E/F\) is Galois.
- \(E/F\) is normal and separable.
- \(E\) is the splitting field of a separable polynomial over \(F\).
- For all elements in \(E\), its irreducible polynomial over \(F\) is separable and splits over \(E\).
Let \(E/F\) be a Galois extension. For all intermediate field \(K\) between \(E\) and \(F\), \(E/K\) is Galois.
Let \(E/F\) be a finite field extension and \(K\) be an intermediate field between \(E\) and \(F\). Then
- If \(E/F\) is normal, then \(E/K\) is normal.
- If \(E/F\) is separable, then \(E/K\) and \(K/F\) are separable.
- If \(E/F\) is Galois, then \(E/K\) is Galois.
\(E/F\) 是 normal extension，代表 \(E\) 含有 \(F[x]\) 中某個不可約多項式 \(f(x)\) 的所有零點。對於 \(K\)，我們可以取同一不可約多項式 \(f(x)\)，也就是說
\[E = F_f \implies E = K_f \implies E/K \text{ is normal.}\]但是 \(K\) 不見得包含所有 \(f(x)\) 的零點，於是 \(K/F\) 不一定 normal！
split: linear factors
separable: no repeated zeros (in algebraic closure)