Cyclotomic polynomial

The $𝑛$ th cyclotomic polynomial is defined to be #m/def/num/alg

Φ 𝑛 (𝑥) : = \prod 1 \leq 𝑚 \leq 𝑛; g c d {𝑚, 𝑛} = 1 (𝑥 - 𝜁 𝑚 𝑛) \in ℤ [𝑥]

is irreducible in $ℚ [𝑥]$ , and has degree given by the Euler totient function $𝜙 (𝑛)$ . Thus this is a minimal polynomial over $ℚ$ for a primitive $𝑛$ th root of unity, and can be used to construct the cyclotomic field $ℚ (𝜁 𝑛)$ .

Proof

By ^P1 $Φ 𝑛$ can be reduced down to a product of prime cyclotomic polynomials which we know have integer coëfficients.

Now suppose towards contradiction that $Φ 𝑛 (𝑥) \in ℤ [𝑥]$ is reducibile. Then its roots $𝜁 𝑚 𝑛$ with $⟨ 𝑚, 𝑛 ⟩ = ⟨ 1 ⟩$ are divided among the factors, so choose a root $𝜁 𝑚 𝑛$ of one irreducible monic factor $𝑓 (𝑥) ∣ Φ 𝑛 (𝑥)$ , so that another root $𝜁 𝑚 𝑝 𝑛$ for a prime $𝑝 ∤ 𝑛$ is not a root of $𝑓 (𝑥)$ . Write

Φ 𝑛 (𝑥) = 𝑓 (𝑥) 𝑔 (𝑥)

where $𝑓 (𝑥), 𝑔 (𝑥) \in ℤ [𝑥]$ are monic. Then $𝑓 (𝑥)$ is the minimal polynomial of $𝜁 𝑚 𝑛$ over $ℚ$ and $𝑔 (𝜁 𝑚 𝑝 𝑛) = 0$ .

It follows that $𝜁 𝑚 𝑛$ is a root of $𝑔 (𝑥 𝑝)$ , and hence $𝑓 (𝑥) ∣ 𝑔 (𝑥 𝑝)$ , whence

𝑔 (𝑥 𝑝) = 𝑓 (𝑥) ℎ (𝑥)

for $ℎ (𝑥) \in ℤ [𝑥]$ . Letting underling denote the projection $ℤ [𝑥] ↠ 𝕂 𝑝 [𝑥]$ , and invoking the Frobenius automorphism we have

𝑔 ―― (𝑥) 𝑝 = 𝑓 ―― (𝑥) ℎ ―― (𝑥)

so $𝑔 ―― (𝑥)$ and $ℎ ―― (𝑥)$ have a nontrivial common factor $ℓ ―― (𝑥)$ in $𝕂 𝑝 [𝑥]$ . It follows

ℓ ―― (𝑥) 2 ∣ 𝑓 ―― (𝑥) 𝑔 ―― (𝑥) = Φ 𝑛 ――― (𝑥)

so $Φ 𝑛 ――― (𝑥)$ has a multiple factor.

This implies that $𝑥 𝑛 - 1 \in 𝕂 𝑝 [𝑥]$ is inseparable. On the other hand, its derivative $𝑛 𝑥 𝑛 - 1 \in 𝕂 𝑝 [𝑥]$ is nonzero (since $𝑝 ∤ 𝑛$ ), contradicting ^P1. Therefore $Φ 𝑛 (𝑥)$ must be irreducible.

Cyclotomic polynomial for a prime power

For the particular case of $𝑛 = 𝑝 ℎ$ we have

Φ 𝑛 (𝑥) = 𝑥 𝑝 ℎ - 1 𝑥 𝑝 ℎ - 1 - 1 = 𝑝 - 1 \sum 𝑗 = 0 𝑥 𝑗 𝑝 ℎ - 1 = 𝑥 (𝑝 - 1) 𝑝 ℎ - 1 + \dots + 𝑥 𝑝 ℎ - 1 + 1

Properties

For all $𝑛 \in ℕ$ ,

𝑥 𝑛 - 1 = \prod 1 \leq 𝑑 ∣ 𝑛 Φ 𝑑 (𝑥)

Proof of 1

If $𝑛 = 𝑑 𝑒$ , then every $𝑑$ th root $𝜁$ of unity is also an $𝑛$ th root of 1, which holds in particular for every primitive $𝑑$ th root $𝜁$ of unity.

On the other hand, every $𝜁 \in 𝜇 𝑛$ generates a subgroup $𝐻 \leq 𝜇 𝑛$ , where if $o r d 𝜁 = 𝑑$ then $𝐻 = 𝜇 𝑑$ . Thus every $𝜁 \in 𝜇 𝑛$ is a primitive $𝑑$ th root of unity for some $𝑑 ∣ 𝑛$ .

Therefore the set of $𝑛$ th roots of unity eauals the union of the sets of primitive $𝑑$ th roots of unity as $𝑑$ ranges over factors of $𝑛$ , whence follows ^P1.

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