Determine the splitting field of $x^4 - 7$ over

Question

Determine the splitting field of $x^4 - 7$ over

(a) $\mathbb{Q}$

(b) $\mathbb{F}_{5}$

(c) $\mathbb{F}_{11}$

For (a): $x^4 - 7 = (x-\sqrt[4]{7})(x+\sqrt[4]{7})(x-i\sqrt[4]{7})(x+i\sqrt[4]{7})$. The splitting field of $x^4 - 7$ is $\mathbb{Q}(\sqrt[4]{7},i)$.

For (b) and (c): I want to determine the splitting field over $\mathbb{F}_{p}$ (for $p \neq 7$, of course). How can I determine this? Is possible?

Answer 1

My algorithm for the fields $\Bbb{F}_p$:

1. Determine the order of $7$ as a root of unity, i.e. its order in the multiplicative group.
2. Determine the orders of the roots of the polynomial in the multiplicative group of the splitting field, call the largest of them $m$ (they may not all be equal, but the others are factors of the largest).
3. Find the smallest exponent $n$ such that $m\mid p^n-1$.
4. The field $\Bbb{F}_{p^n}$ is the smallest containing $m$th roots of unity. As it contains all of them it must be the splitting field.