What is wrong with this proof that a polynomial is solvable by radicals?

Question

We consider $$x^5 - x + 5 \in \mathbb{Q}[x].$$

I'm pretty sure that this polynomial is NOT solvable by radicals since its Galois group is isomorphic to $S_5$ (see my question from yesterday).

However, my teacher is pretty sure that it IS solvable by radicals, and he proved it by seeing that its Galois group is isomorphic to $D_5$ (and I think it's not).

He considers the roots of the polynomial $\alpha_1, \alpha_2, \overline{\alpha_2}, \alpha_3, \overline{\alpha_3}$, with $\alpha_1 \in \mathbb{R}, \alpha_2, \alpha_3 \in \mathbb{C}\setminus\mathbb{R}$. These roots form a regular pentagon, and since two pairs of roots are conjugate, we can consider the rotation $a = (1, 2, 3, 4, 5)$ and the permutation $b = (2, 5)(3, 4)$, which generate $$D_5 = \left\langle a, b \mid a^5 = 1 = b^2, ba = a^{-1}b \right\rangle.$$

So then he saw that $D_5$ is solvable by radicals and deduced that the polynomial is solvable by radicals too.

I don't really understand at all what he does, so I am not able to say what is wrong with it… Could you help me? Thanks!

Answer 1

No, the pentagon formed by the roots of

$$x^5-x+5=0 \tag{1}$$

is neither regular nor centered in $0$ (one can be fooled by the fact that the axes haven't necessarily the same length if it is not explicitly asked).

One can check it on the approximate values of the roots, which are :

$$\begin{cases}r_1&\approx&-1.4519\\r_2&\approx&-0.36823+1.3592i\\r_3&=&\overline{r_2}\\r_4&\approx&1.0942 + 0.73443i\\r_5&=&\overline{r_4}\end{cases},$$

giving very different sidelengths, for example $\approx 1.73233$ between $r_1$ and $r_2$, and $\approx 2.18836$ between $r_4$ and $\overline{r_4}$.

It is not centered in $0$ because the norms of the $r_k, k=1,\cdots 5$ are :

$$\approx 1.4519, 1.4082, 1.4082, 1.3178, 1.3178$$

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Edit : A rigorous proof, by contradiction, can be given. Let us assume that the pentagon is regular and centered in the origin : this would mean that the roots, now with indices $k=0,\cdots 4$ would be :

$$r_k=re^{ik \tfrac{2 \pi}{5}}$$

where $r=r_0\ne 0$ is the unique real root.

Their product, using one of the Viète's formulas, would give rise to :

$$r^5\underbrace{e^{i (0+1+2+3+4) \tfrac{2\pi}{5}}}_{= \ 1}=-5\tag{2}$$

Plugging (2) into the following relationship (true because $r$ is a root of equation (1)) :

$$r^5-r+5=0$$

leads to $r=0$ which is impossible.

As I know you are interested by Sage, here is the little program which has produced in particular the graphical representation :

CC=ComplexField(20)
p=x^5-x+5;
s=p.roots(ring=CC);
L=[];M=[];
for k in range(len(s)):
    m=s[k][0];
    L.append([m.real(),m.imag()]);
    M.append(abs(m));
print(L);
print(M);
Q=points(L,size=50);
show(Q,aspect_ratio=1);