What does an idiosyncratic solution in radicals means? I read that Galois theory asserts that some equations of at least degree 5 don't have an idiosyncratic solution in radicals. So, what does this statement actually means?

Idiosyncratic means unique to an individual.

This is a quote from Wikipedia


Solving quintic equations in terms of radicals was a major problem
in algebra, from the 16th century, when cubic and quartic equations
were solved, until the first half of the 19th century, when the
impossibility of such a general solution was proved
(Abel–Ruffini theorem).


All $n^\text{th}$-degree polynomials with rational coefficients have $n$ roots. The question was, can these roots be written as radical expressions. The hope was that, if the expression was complicated enough, you could. It turned out that, for $n \ge 5$, sometimes you can't.