Consider $a$ as a transcendental over $\mathbb C$. Then the $x_i$'s generate a Galois extension of $\mathbb C(a)$ of degree $n$ with Galois group the symmetric group $S_n$. Thus once $n\ge5$, you cannot solve for the $x_i$'s in terms of $a$ by radicals.

Consider $a$ as a transcendental over $\mathbb C$. Then each $x_i$ generates an extension of $\mathbb C(a)$ of degree $n$ whose Galois closure has Galois group the symmetric group $S_n$. Thus once $n\ge5$, you cannot solve for the $x_i$'s in terms of $a$ by radicals.

Consider $a$ as a transcendental over $\mathbb C$. Then the $x_i$'s generate a Galois extension of $\mathbb C(a)$ of degree $n$, thus once $n\ge5$, you cannot solve for the $x_i$'s in terms of $a$ by radicals.

Consider $a$ as a transcendental over $\mathbb C$. Then the $x_i$'s generate a Galois extension of $\mathbb C(a)$ of degree $n$ with Galois group the symmetric group $S_n$. Thus once $n\ge5$, you cannot solve for the $x_i$'s in terms of $a$ by radicals.