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假設現在已經知道 Galois group \(G\) 的某個 subgroup \(H\)，如何找出對應的 subfield \(E^H\)？法一：設 \(H=\langle a\rangle\)。根據定義，對於所有 \(x\in E^H\)，\(ax = x\)。由於 \(x\) 也在 \(E\) 中，我們可以將 \(x\) 表示為 \(c_0 + c_1\alpha + \cdots\) 的形式，再比較 \(ax\) 和 \(x\) 的各項係數就好！底下再提出另一種方法
Let \(E/F\) be a field extension with Galois group \(G\), and \(K\) be an intermediate field. Define the trace of \(x\in E\) on \(E/K\):
\[\text{tr}(x) = \sum_{h\in H}h(x),\]where \(H\le G\).
Proof (image is contained in \(E^H\))
For all \(h'\in H\),
\[\begin{align*} h'(\text{tr}(x)) &= h'\big(\sum_{h\in H}h(x) \big) \\ &= \big(\sum_{h\in H}h'h(x) \big) \tag*{since $h'$ is an automorphism} \\ &= \sum_{h\in H}h(x) \tag*{left multiplication: permute} \\ &= \text{tr}(x). \tag*{$\blacksquare$} \end{align*}\]左乘是個 bijective mapping！（試著證明）
Let \(\beta\) be a basis of \(E\) over \(F\). Then \(E^H = \text{tr}(E)\) is spanned by \(\text{tr}(\beta)\).
然後再把 \(\beta\) 中的元素代入 \(\text{tr}\)，解 \(E^H\) 的 basis！
待補齊…
Define the norm of \(x\in E\) on \(E/F\):
\[N_{E/F}(x) := \prod_{g\in G}g(x).\]
Let \(E/F\) be a finite Galois extension with the Galois group \(G\). For all \(\alpha\in E\), we have \(N_{E/F}(\alpha) = \det(L_\alpha)\) and \(\text{tr}_{E/F}(\alpha) = \text{tr}(L_\alpha)\).
Proof hint
先明確寫出當 \(E = F(\alpha)\) 的 trace 和 norm 看看。