I'm reading this document: http://www-groups.mcs.st-and.ac.uk/~neunhoef/Teaching/ff/ffchap4.pdf
Given a finite field $F_q$, a cyclotomic polynomial $Q_n(x)$ with $gcd(q, n) = 1$, and $d = ord_n(q)$.
I want to proof that after factorizing the $Q_n(x)$ to irreducible polynomials, all of these factors have degree d.
I have found this document http://www-groups.mcs.st-and.ac.uk/~neunhoef/Teaching/ff/ffchap4.pdf.
and i don't quite understand the proof of Theorem 8.12. It says:
Now let K be the finite field $F_q$, assume gcd(q, n) = 1, such that primitive nth roots of unity over $F_q$ exist. Let η be one of them. Then
η ∈ $F_{q^k}$ ⇔ $η^{q^k}$ = η ⇔ $q^k$ ≡ 1 mod n.
The smallest positive integer for which this holds is k = d, so η is in $F_{q^d}$ but not in any proper subfield. Thus the minimal polynomial of η over $F_q$ has degree d. Since η was an arbitrary root of $Q_n(x)$, the result follows, because we can successively divide by the minimal polynomials of the roots of $Q_n(x)$.
I dont get the Step
η is in $F_{q^d}$ but not in any proper subfield.
to
Thus the minimal polynomial of η over $F_q$ has degree d
I feel like this is something elementary that i'm missing.