An introduction to the modern theory of equations, by Florian Cajori.

CYCLIC EQUATIONS 209
G3(3). Here this index is 3. As n is prime, n-=3. This makes t2(x)=-(p),
where p is a root of the normal cubic. Hence the roots of xn - a = 0 are
the conjugate values of X, ~ 136, and all of them lie in the normal domain
0(p). Now, if one root of a normal equation is real, all its roots are real.
Therefore, all the roots of -n - a = 0, being functions in 2 of p would
have to be real. But this cannot be, when n = 3. Thus, the assumption
that our cubic can be solved by real radicals of prime order leads to an
absurdity.
Nor would the solution be possible by real radicals of composite order,
such as ~/a, where n = pq, a composite number; for, in that case we can
write -/ —V and we can adjoin in succession the radicals of prime order
y-a and /y. But, as has just been shown, such adjunctions do not
render the normal cubic reducible.