Cyclotomic field

Remarks

The minimal polynomial of $\zeta_n$ over $\Q$ is the cyclotomic polynomial $\Phi_n$, which is irreducible over $\Q$ by the irreducibility of cyclotomic polynomials. So $[\Q(\zeta_n) : \Q] = \varphi(n)$, where $\varphi$ is Euler's totient function, and $\Q(\zeta_n)$ is an algebraic number field. Example: for $n = 4$, $\zeta_4 = i$ and $\Phi_4 = x^2 + 1$, so $\Q(\zeta_4) = \Q(i)$ has degree $\varphi(4) = 2$. Non-example: $\Q(\sqrt[3]{2})$ is not a cyclotomic field: every cyclotomic field is a Galois extension of $\Q$, but $\Q(\sqrt[3]{2}) \subset \R$ does not contain the non-real roots of $x^3 - 2$. The Galois group of $\Q(\zeta_n)/\Q$ is isomorphic to $(\Z/n\Z)^{\times}$, with $a$ acting by $\zeta_n \mapsto \zeta_n^a$. The ring of integers of $\Q(\zeta_n)$ is $\Z[\zeta_n]$. For odd $n$, $\Q(\zeta_{2n}) = \Q(\zeta_n)$, because $-\zeta_n$ is a primitive $2n$-th root of unity.