I'd be grateful for a good reference on this, it feels like a classic subject yet I couldn't find much about it.

Polynomials in one variable of the form $x^n+a_{n-1}x^{n-1}+\dots +a_1 x+a_0$ can be transformed into simpler expressions. For instance it is apparently well-known that the Tschirnhaus transformation allows to bring *any* quintic into so-called Bring-Jerrard form $x^5+ax+b$, while for degree 6 one needs at least three coefficents $x^6+ax^2+bx+c$.

Is there a name for such "generalized Bring-Jerrard form", and what is known about it? In particular there is a cryptic footnote of Arnold (page 3 of this lecture) where he says roughly that the degrees for which more coefficients are needed occur along "a rather strange infinite sequence": could someone please describe what those degrees are (I had a look at the OEIS but I believe that sequence is different from Hamilton numbers, and couldn't find a relevant one).