How can I see if a polynomial is solvable by radicals or not?

Question

We consider $$x^5 + 5x^4 + 10x^3 + 10x^2 + 5x - 2 \in \mathbb{Q}[x]$$ and $$x^5 - x + 5 \in \mathbb{Q}[x].$$ I want to see if they are solvable by radicals or not.

I know that firstly I should determine the splitting field of our polynomials and the Galois group of it. However, I can't think of how to do this...

Could you help me, please? Thanks!!

Answer 1

For the second polynomial

$$p(x):=x^5-x+5$$

Theoretical side : there are results on Galois groups of such monic polynomials with $3$ terms which can be found here. If we consider its Lemma 9, as $1+1<5$, $p(x)$ is irreducible. Moreover, the discriminant of $p$ (see below) is $1952869=239 \times 8171$ without square factors. Therefore, Theorem 6 of the same paper can be applied : the Galois group of $p$ is the symmetric group $S_5$.

Practical side : here is a Sagemath program :

p=QQ['x'](x^5-x+5)
d=p.discriminant();show(d);show(factor(d))
gal=p.galois_group();
n=gal.order();show(n)
gal.is_isomorphic(SymmetricGroup(5))

giving the answers :

Discriminant : $d=1952869=239 \times 8171$

Galois group order : $n=120=5!$

With the confirmation that the boolean given by the fourth instruction is "True".