Question

Show that the polynomial
X5=9x + 3
Is not solvable by radicals over

Answer 1

For a polynomial of degree 5, we simply consider the derivative and determine the number of real and complex roots from there.

$p(x) =x^{5}-9x-3$

$f$ is an irreducible polynomial of prime degree $p$ since the polynomial cannot be factored into nontrivial polynomials over the same field, then the Galois group of $f$ is the full symmetric group Sp.

In this case, $p$ =5 and S₅ is not solvable