Cyclotomic field

A cyclotomic field $𝐾 𝑚 : = ℚ (𝜁 𝑚)$ is a number field obtained by adjoining a primitive $𝑚$ th root of unity, #m/def/num/alg i.e. $𝜁 𝑚 = e 2 𝜋 𝑖 / 𝑚$ , or equivalently, the splitting field of the separable polynomial

𝑥 𝑛 - 1 .

It follows that $𝐾 𝑚 : ℚ$ is a Finite Galois extension, with $G a l (𝐾 𝑚 : ℚ) = ℤ \times 𝑚$ and degree $𝜙 (𝑚)$ given by the Euler totient function. The defining minimal polynomial of such a field is the so-called Cyclotomic polynomial.

Its ring of integers are the Cyclotomic integers $ℤ [𝜁 𝑚]$ .
See Group of roots of unity

This is especially well-behaved when $𝑚$ is a prime power, see Prime power cyclotomic field.

Properties

The discriminant $Δ 𝐾 𝑚 : ℚ (𝜁 𝑚)$ divides $𝑚 𝜙 (𝑚)$ .¹

Proof of 1.

Since $Φ 𝑚 (𝑥) ∣ (𝑥 𝑚 - 1)$ , it follows $𝑥 𝑚 - 1 = Φ 𝑚 (𝑥) 𝑔 (𝑥)$ for some $𝑔 (𝑥) \in ℤ [𝑥]$ . Then

𝑚 𝜁 𝑚 - 1 𝑚 = Φ' 𝑚 (𝜁 𝑚) 𝑔 (𝜁 𝑚)

since $Φ 𝑚 (𝜁 𝑚) = 0$ . Taking the norm of both sides

𝑚 𝜙 (𝑚) = N 𝐾 𝑚 : ℚ (Φ' 𝑚 (𝜁 𝑚)) N 𝐾 𝑚 : ℚ (𝑔 (𝜁 𝑚)) = \pm Δ 𝐾 𝑚 : ℚ (𝜁 𝑚) N 𝐾 𝑚 : ℚ (𝑔 (𝜁 𝑚))

where $N 𝐾 𝑚 : ℚ (𝑔 (𝜁 𝑚)) \in ℤ$ , proving ^P1.

#state/develop | #lang/en | #SemBr

2022. Algebraic number theory course notes, §2.4.1, p. 47 ↩

Cyclotomic field

Properties

Footnotes