TITLE: Determine the splitting field of $x^4 - 7$ over
QUESTION [6 upvotes]: 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?

REPLY [6 votes]: My algorithm for the fields $\Bbb{F}_p$: 

Determine the order of $7$ as a root of unity, i.e. its order in
the multiplicative group. 
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). 
Find the smallest exponent
$n$ such that $m\mid p^n-1$. 
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.