I'm having trouble in understanding "the form" of a splitting field.
The problem is: Construct the splitting field over $\mathbb{Q}$ of the following polynomials.
One of the polynomials is $x^{4}-3$
So, the roots of this polynomial are $\pm \sqrt[4]{3}$ and $\pm \sqrt[4]{3}i$. Therefore, the splitting field over $\mathbb{Q}$ of this polynomial is $\mathbb{Q}[\sqrt[4]{3},i]$, right?
But what is the form of these elements?
Like, we have that $\mathbb{Q}[\sqrt{2}]=\{a_1+a_2\sqrt{2};a_1,a_2 \in \mathbb{Q}\}$. So, $\mathbb{Q}[\sqrt[4]{3},i]$ = ?
Best Answer
Remember that $\;Q(\sqrt[4]3\,,\,i)\;$ is a vector space over $\;Q\;$ of dimension $\;4\cdot 2=8\;$ , and it has a very nice basis (putting $\;w:=\sqrt[4]3\;$ for simplicity, we get):
$\;\{1\,,\,w\,,\,w^2\,,\,w^3\,,\,i\,,\,wi\,,\,w^2i\,,\,w^3i\}\;$ , so you can conveniently write any element in the above field in the form
$$a+bw+cw^2+dw^3+ewi+fw^2i+gw^3i\;,\;\;a,b,c,d,e,f,g\in\Bbb Q$$