Galois group
Cyclotomic extensions

Another useful class of examples comes from the splitting fields of cyclotomic polynomials. These are polynomials Φ n {\displaystyle \Phi _{n}} defined as

whose degree is ϕ (n) {\displaystyle \phi (n)}, Euler's totient function at n {\displaystyle n}. Then, the splitting field over Q {\displaystyle \mathbb {Q} } is Q (ζ n) {\displaystyle \mathbb {Q} (\zeta _{n})} and has automorphisms σ a {\displaystyle \sigma _{a}} sending ζ n ↦ ζ n a {\displaystyle \zeta _{n}\mapsto \zeta _{n}^{a}} for 1 ≤ a < n {\displaystyle 1\leq a<n} relatively prime to n {\displaystyle n}. Since the degree of the field is equal to the degree of the polynomial, these automorphisms generate the Galois group. If n = p 1 a 1 ⋯ p k a k, {\displaystyle n=p_{1}^{a_{1}}\cdots p_{k}^{a_{k}},} then

If n {\displaystyle n} is a prime p {\displaystyle p}, then a corollary of this is

In fact, any finite abelian group can be found as the Galois group of some subfield of a cyclotomic field extension by the Kronecker–Weber theorem.
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