Let $k$ be a field of characteristic $0$ and let $a,b,c,d,e \in k$. Then is the polynomial $f(x)=ax^8+bx^6+cx^4+dx^2+e$ solvable by radicals over $k$?
In other words, is the Galois group of $f$ over $k$ solvable?
2026-03-27 15:20:10.1774624810
$\operatorname{char} k=0$, $a,b,c,d,e \in k$ , then is the polynomial $f(x)=ax^8+bx^6+cx^4+dx^2+e$ solvable by radicals over $k$?
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Hint: Is $g(x)=ax^4+bx^3+cx^2+dx+e$ solvable by radicals?