Hi Paul,
thank you very much for this nice notebook. This really makes the article in the quanta magazine twice as valuable.
I only want to add one more sentence, which makes the difference between the two polynomials more transparent, hopefully.
For your function f(x) ( in the article it is denoted with g(x)) the expression of the roots
$(a-b)(b-c)(c-a)=\pm 7$ is rational and therefore breaks half of the symmetries.
For the other function g(x), the expression of the roots
$(r-s)(s-t)(t-s)\notin {\mathbb Q}$ is irrational and therefore it cannot be used to reduce the number of symmetries.
$g(x)$ ends up with the full symmetric group as Galois group.
Finally, when you look at the radical expressions for the roots you can convince yourself, that the solutions for
$f(x)$ are simpler than for
$g(x)$. While the solutions for
$g(x)$ contain nested roots the solutions for
$f(x)$ don't.
