More of a comment than an answer, but there are exceptions for $n = 17$ and $n = 145$, where the Galois group is simple of order 168. This is all the exceptions for $|n| < 10^5$2\times10^5$. 1 More of a comment than an answer, but there are exceptions for$n = 17$and$n = 145$, where the Galois group is simple of order 168. This is all the exceptions for$|n| < 10^5\$.