Cyclotomic polynomial

The th cyclotomic polynomial is defined to be alg

is irreducible in , and has degree given by the Euler totient function . Thus this is a minimal polynomial over for a primitive th root of unity, and can be used to construct the cyclotomic field .

Proof

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

Now suppose towards contradiction that is reducibile. Then its roots with are divided among the factors, so choose a root of one irreducible monic factor , so that another root for a prime is not a root of . Write

where are monic. Then is the minimal polynomial of over and .

It follows that is a root of , and hence , whence

for . Letting underling denote the projection , and invoking the Frobenius automorphism we have

so and have a nontrivial common factor in . It follows

so has a multiple factor.

This implies that is inseparable. On the other hand, its derivative is nonzero (since ), contradicting ^P1. Therefore must be irreducible.

Cyclotomic polynomial for a prime power

For the particular case of we have

Properties

For all ,

\begin{align*} x^n-1 = \prod_{1\leq d \mid n} \Phi_{d}(x) \end{align*}

You can't use 'macro parameter character #' in math mode^P1 > [!check]- Proof of 1 > > If $n = de$, then every $d$th root $\zeta$ of unity is also an $n$th root of 1, > which holds in particular for every _primitive_ $d$th root $\zeta$ of unity. > > On the other hand, every $\zeta \in \mu_{n}$ generates a subgroup $H \leq \mu_{n}$, > where if $\opn{ord} \zeta = d$ then $H = \mu_{d}$. > Thus every $\zeta \in \mu_{n}$ is a primitive $d$th root of unity for some $d \mid n$. > > 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|^P1]]. <span class="QED"/> # --- #state/develop | #lang/en | #SemBr

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Table of Contents

Cyclotomic polynomial

Cyclotomic polynomial for a prime power

Properties

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