Is the following polynomial solvable in radicals?

Question

Is the polynomial $x^8-x^6+2x^4-6x^2+1$ solvable in radicals over $\mathbb{Q}$? I am unsure how to solve this. I don't know how to compute the Galois group, and the discriminant seems much to hard to compute. Any suggestions will be appreciated.

Answer 1

Yes it is solvable in radicals. As all exponents are even, you can substitute $x^2=t$ to reduce to a degree $4$ polynomial. Every degree $4$ polynomial is solvable in radicals.

Then just take the square-roots of the solutions of the degree $4$ polynomial.

Answer 2

It is certainly solvable in radicals, since it is merely a quartic polynomial in the variable $x^2$, which is solvable in radicals. Then take the square roots of those four solutions.