Let $G$ be a multiplicative group modulo 16, i.e, $(\mathbb{Z}/16\mathbb{Z})^\times$.
So we know $$ G = \{1,3,5,7,9,11,13,15\}. $$ Now, we have a cyclic group $\langle 3 \rangle = \{1, 3, 9, 11\}$ as a subgroup.
So we can have the quotient group that consists of 2 cosets: $$ Z = G/\langle 3 \rangle = \{\{1, 3, 9, 11\}, \{5, 7, 13 ,15\}\}. $$ Let denote $H_1 = \{1, 3, 9, 11\}$ and $H_2 = \{5, 7, 13, 15\}$.
On the other hand, we have $(X^8 + 1) \equiv f_1(X)f_2(X) \mod 3 $.
And the following holds $\mathcal{Gal}(\mathbb{Q}[\zeta]/\mathbb{Q}) \cong G$.
Question 1: Is it possible to determine $f_1(X)$ and $f_2(X)$ with $H_1$ and $H_2$ respectively?? Let $f_1(X) = a_4 X^4 + a_3 X^3 + a_2 X^2 + a_1 X + a_0$ and $f_2(X) = b_4 X^4 + b_3 X^3 + b_2 X^2 + b_1 X + b_0$. Is there any relationship between coefficients of $f_1(X)$, $f_2(X)$ and $H_1, H_2$?
Question 2: If some permutation $\sigma_i$ for $i \in G$ in the Galois group is applied, is there any $i$ that swaps $f_1(X)$ and $f_2(X)$??