Finding the Number of Subfields of the Splitting Field of $x^{35}-1$ over $\mathbb{F}_8$

Question

Let $E$ be the splitting field of $x^{35}-1$ over the field $\mathbb{F}_8$. Determine $|E|$ and the number of subfields of $E$.

Attempt: I am confident that I computed $|E|$ correctly, but I am not sure if I have correctly determined the number of subfields of $E$.

Since $\mathbb{F}_8=\mathbb{F}_{2^3}$, $\mathbb{F}_8$ has characteristic $2$. Denote by $I_n(x)$ the $n$th cyclotomic polynomial, and denote by $\varphi$ Euler's Totient. Then $$x^{35}-1=I_1(x)I_5(x)I_7(x)I_{35}(x)$$ where $I_1(x)=x-1, I_5(x)=x^4+x^3+x^2+x+1, I_7(x)=x^6+x^5+\cdots +x+1$, and $I_{35}(x)$ can be computed by recursion but I don't believe the formula is needed. The root $1$ of $I_1$ gives us no new elements; the roots of $I_5$ give $\varphi(5)=4$ nonreal elements (which cannot be in $\mathbb{F}_8$ since $\mathbb{F}_8$ has characteristic $2$); the roots of $I_7$ give $\varphi(7)=6$ new elements; the roots of $I_{35}$ give $\varphi(35)=\varphi(5)\varphi(7)=24$ new roots. No roots were counted twice since the $I_n$'s are distinct irreducible polynomials. So $|E|=8+4+6+24=42$, and there are $5$ subfields (including $\mathbb{F}_8, E$) where the 3 nontrivial subfields are given by including the roots of $I_5, I_7, I_{35}$. Is the last statement true, and can it be made more rigorous using Galois theory?

Answer 1

Hints:

• All the non-zero elements of $\Bbb{F}_8$ are seventh roots of unity, all but one of them of order seven.
• The smallest field of characteristic two that contains a primitive fifth root of unity if $\Bbb{F}_{16}$. Can you see why that is the case? Think cyclic groups!
• If a field contains a primitive fifth root of unity and a primitive seventh root of unity, then it also contains a primitive $35$th root of unity. Why?
• In view of the previous bullet $E$ is the smallest field that contains both $\Bbb{F}_8$ and $\Bbb{F}_{16}$ (as well as the field you are working over, but that happens to be in the list already). Review what you know about inclusions of finite fields, and determine $E$.