Degree of $\mathbb{F}_p(\zeta_r)$ over $\mathbb{F}_p$

Question

We know that $r^{th}$ cyclotomic polynomial splits as equidegree factors over $\mathbb{F}_p$ of degree $ord_r(p)$. But what can we say about Degree of $\mathbb{F}_p(\zeta_r)$ over $\mathbb{F}_p$ ??

Answer 1

By definition $\mathbb{F}_q(\zeta)$ is the smallest finite field $\mathbb{F}_{q^{\large e}}$ which contains an $n$th root of unity. The cyclic group $\mathbb{F}_{q^{\large e}}^\times$ of order $q^e-1$ contains an element of order $n$ if and only if $n\mid (q^e-1)$, so we need to find the smallest $q^e-1$ divisible by $n$, or equivalently the smallest $e$ for which $q^e\equiv 1$ mod $n$. This is precisely the multiplicative order of $q$ within $(\mathbb{Z}/n\mathbb{Z})^\times$.