Galois theory of cyclotomic extensions

Let \(\zeta_n\) be a primitive \(n\)th root of unity. The cyclotomic polynomial is \[ \Phi_n(x)=\prod_{(a,n)=1}(x-\zeta_n^a). \] It lies in \(\mathbb Z[x]\), is monic, has degree \(\varphi(n)\), and is irreducible over \(\mathbb Q\). Hence \[ [\mathbb Q(\zeta_n):\mathbb Q]=\varphi(n). \]

The field \(\mathbb Q(\zeta_n)\) is the splitting field of \(\Phi_n\), so it is Galois over \(\mathbb Q\). Every automorphism is determined by the image of \(\zeta_n\), and that image must be another primitive \(n\)th root. Thus \[ \operatorname{Gal}(\mathbb Q(\zeta_n)/\mathbb Q) \longrightarrow (\mathbb Z/n\mathbb Z)^\times, \qquad \sigma\longmapsto a \quad\text{when }\sigma(\zeta_n)=\zeta_n^a, \] is an injective homomorphism. Both groups have order \(\varphi(n)\), so it is an isomorphism: \[ \operatorname{Gal}(\mathbb Q(\zeta_n)/\mathbb Q) \cong(\mathbb Z/n\mathbb Z)^\times. \] In particular the extension is abelian.

By Galois correspondence, intermediate fields are in inclusion-reversing bijection with subgroups of \((\mathbb Z/n\mathbb Z)^\times\).

For \(n>2\), complex conjugation corresponds to \(-1\in(\mathbb Z/n\mathbb Z)^\times\) and has order \(2\). Its fixed field is the maximal real subfield \[ \mathbb Q(\zeta_n)^+ =\mathbb Q(\zeta_n+\zeta_n^{-1}), \] with \[ [\mathbb Q(\zeta_n):\mathbb Q(\zeta_n)^+]=2. \] For \(n=1,2\), the cyclotomic field is already \(\mathbb Q\), so this index is \(1\).

For an odd prime \(p\), \((\mathbb Z/p\mathbb Z)^\times\) is cyclic, hence has a unique subgroup of index \(2\); correspondingly \(\mathbb Q(\zeta_p)\) has a unique quadratic subfield, \[ \mathbb Q\!\left(\sqrt{(-1)^{(p-1)/2}p}\right). \] Finally, the Kronecker–Weber theorem states that every finite abelian extension of \(\mathbb Q\) is contained in some cyclotomic field.