Table of Content
Too many things to write. Here are only some important notes.
Any subextension of a cyclotomic extension \(\Bbb Q(\zeta_n)/\Bbb Q\) is Galois.
Proof
We have \(\text{Gal}(\Bbb Q(\zeta_n)/\Bbb Q) \cong \Bbb Z_n^\times\) and \(\Bbb Z_n^\times\) is abelian. Since every subgroup of an abelian group is normal, by the Galois theory, any subextension of \(\Bbb Q(\zeta_n)/\Bbb Q\) is normal, and hence Galois. ■
Let \(p\) be a prime. Then any transitive subgroup of \(S_p\) has a \(p\)-cycle.
Proof
Let \(H\) be a transitive subgroup of \(S_p\), and let \(H\) act on \(\{1, 2, \cdots, p\}\). By the orbit-stabilizer theorem, we have
\[\vert H\vert = \vert H_x\vert \cdot \vert \text{Stab}_H(x)\vert = pq,\]for some positive integer \(q\). Then by Cauchy’s theorem, \(H\) has an element of order \(p\), which implies that \(H\) has a \(p\)-cycle. ■
Since \(H\) is transitive, \(\vert H_x\vert = p\).
If \(f(x)\) is irreducible over \(\Bbb Q\) of some prime degree \(p\), then \(G = \text{Gal}(\Bbb{Q}_f/\Bbb Q)\) contains a \(p\)-cycle.
Proof
Since \(f(x)\) is irreducible, \(\deg(f) = p \mid \vert G\vert\). By Cauchy’s theorem, \(G\) has an element of order \(p\). ■
計算 Galois group \(G\) 的時候，只要給定的不可約多項式 \(f(x)\) 的次數 \(n\) 是質數，\(G\) 就一定有 \(n\)-cycle！但若 \(n\) 不是質數，就得找某質數 \(p\)，使得 \(f(x)\) 在 \(\Bbb{Z}_p\) 下是不可約的（然後 by Dedkind’s theorem）。
For all positive integers \(n\), there exists an integer polynomial \(f(x)\) of degree \(n\) such that the Galois group of its splitting field \(\Bbb{Q}_f\) over \(\Bbb Q\) is isomorphic to \(S_n\).
For every finite group \(G\), there exists a Galois extension \(E/F\) such that \(\text{Gal}(E/F) \cong G\).
Proof
Suppose \(\vert G\vert = n\). By Hilbert’s theorem, there exists an integer polynomial \(f(x)\) of degree \(n\) such that the Galois group of its splitting field \(\Bbb{Q}_f\) over \(\Bbb Q\) is isomorphic to \(S_n\). Let \(E = \Bbb{Q}_f\). By Cayley’s theorem, \(G\) is isomorphic to a subgroup of \(S_n\). Then, by the fundamental theorem of Galois theory, there exists a subextension of \(E\), \(E^G\), such that \(\text{Gal}(E/E^G) \cong G\). ■