What do you mean by splitting field? Find the splitting field of $x^5 -1$

Question

Okay, first of all sorry for a stupid question. Now what I know is -

Splitting Field: Let $K$ be a field and let $f(x) = a_0 + a_1x + a_2 x^2+\cdots + a_n x^n$ be a polynomial in $K[x]$ of degree $n>0$. An extension field $F$ of $K$ is called a splitting field for $f(x)$ over $K$ if there exist elements $r_1,r_2,\cdots,r_n \in F$ such that
$(i)$ $f(x) = a_n (x-r_1) (x-r_2)\cdots (x-r_n)$, and
$(ii)$ $F = K(r_1,r_2,\cdots,r_n)$

There is another concept: For $f(x)\in F[X]$, which is a polynomial of degree $n\ge 1$, there exist an extension $E$ of $F$ of degree at most $n!$ in which $f(x)$ has $n$ roots.

Let's take an example as $x^4+1$, then $$x^4+1 = (x^4+2x^2+1)-2x^2 = (x^2+1)^2 - (\sqrt{2} x)^2 = (x^2 + 1 + \sqrt2 x)(x^2 + 1- \sqrt{2} x)$$

Let $r$ be the root of $(x^2 + 1- \sqrt2x)$, then $r = -\dfrac{\sqrt2 \pm \sqrt{2-4}}2 = \dfrac{\sqrt2(\iota-1)}2, \dfrac{-\sqrt2(\iota+1)}2 $

So, $\Bbb{Q}(\sqrt2,\iota)$ is the smallest field containing $\dfrac{\sqrt2(\iota-1)}2$
So it's the splitting field right?
$[\Bbb{Q}(\sqrt2,\iota):\Bbb{Q}] = [\Bbb{Q}(\sqrt2,\iota):\Bbb{Q}(\sqrt2)][\Bbb{Q}(\sqrt2,\iota):\Bbb{Q}] = 2\times 2 = 4$
( Because $\sqrt2$ satisfies an irreducible polynomial $x^2-2$ over $\Bbb{Q}$ hence $[\Bbb{Q}(\sqrt2):\Bbb{Q}] = 2$. Similarly $\iota$ satisfies an irreducible polynomial $x^2+1$ over $\Bbb{Q}$, therefore it satisfies $x^2+1$ over $\Bbb{Q}(\sqrt2)$ also hence $[\Bbb{Q}(\sqrt2,\iota):\Bbb{Q}(\sqrt2)]=2$ )
Is my explanation correct?

So first we need to find the factors of $x^5 - 1$.
This equation can be simplified to $$x^5-1= (x^4 + x^3 + x^2 + x + 1)(x - 1)$$ And now I am stuck here. I will update the solution if I'm able to get the roots.
Thanks.

Answer 1

First off, we need a ground field. I'll assume that this is $\Bbb Q$, but it might as well be $\Bbb R$, or a finite field, or any of a number of other options.

The splitting field of $x^5-1$ over $\Bbb Q$ of the smallest field which

1. Contains $\Bbb Q$
2. Contains all roots of $x^5-1$

You are allowed to use things you know from complex analysis to find these roots of $x^5-1$, if you so desire. Once you've found them, say $x_1,x_2,x_3,x_4,x_5$, the splitting field is simply the extension of $\Bbb Q$ by these roots: $\Bbb Q(x_1,x_2,x_3,x_4,x_5)$. If this expression can be simplified (for instance, if some of the $x_i$ are superfluous), then I suggest you simplify it.