In the following problems, let $K$ be the splitting field of $f(x)$ over $F$. Determine Gal$(K/ F)$ and find all the intermediate subfields of $K/F$.
(a) $F = \mathbb{Q}$ and $f (x) = x^4 - 7$.
(b) $F = \mathbb{F_5}$ and $f(x) = x^4 - 7$.
(c) $F = \mathbb{Q}$ and $f (x) = x^5 - 2$.
(d) $F = \mathbb{F_2}$ and $f(x) = x^6 + 1$.
(e) $F = \mathbb{Q}$ and $f(x) = x^8 - 1$.
My attempt: (a) $f (x) = x^4 - 7=(x^2-\sqrt{7})(x^2+\sqrt{7})=(x-\sqrt[4]{7})(x+\sqrt[4]{7})(x-i\sqrt[4]{7})(x+i\sqrt[4]{7})$ so $K=\mathbb{Q}(i,\sqrt[4]{7})$ so $|\text{Gal}(K/F)|=8$ so $\text{Gal}(K/L)\cong \mathbb{Z}_2\times \mathbb{Z}_4$ so $\mathbb{Q}(i), \mathbb{Q}(\sqrt[4]{7}), \mathbb{Q}(i\sqrt[4]{7})$ are intermediate field.
This is OK? Any suggestions to solve the rest?