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Let $x_1,\dots ,x_5$ be algebraically independent indeterminates and set $a_i=e_i(x_1,\dots ,x_5)$, the elementary symmetric polynomials. ThenThe Fundamental theorem of symmetric polynomials identifies the fixed field of the natural $S_5$ action on $\mathbb{Q}(x_1,\dots ,x_5)$ with $\mathbb{Q}(a_1,\dots ,a_5)=K$. Every permutation of the roots therefore gives a $K$-automorphism, and the Galois group of a polynomial isA characteristic-zero polynomial is solvable by radicals exactly when its Galois group is a solvable group. Part (c) shows that $S_5$ is not solvable, so

In the strict tower convention, a radical extension is a field obtained through finitely many adjunctions $a_i^{n_i}\in K_{i-1}$, $K_i=K_{i-1}(a_i)$. Some texts instead use the term for any subfield of such a tower. This distinction matters: solvability by radicals implies containment in a tower, and need not give a radical tower entirely inside the original field.