What is the highest value of $n$ be to reduce a general polynomial $a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0=0$ to $y^n+py+q=0 ?$

Question

We know that the following facts.

The quadratic equation $ax^2+bx+c=0$ can be reduce to $y^2+p=0$
The cubic equation $ax^3+bx^2+cx+d=0$ can be reduce to $y^3+py+q=0$
The quartic equation $ax^4+bx^3+cx^2+dx+e=0$ can be reduce to $y^4+py+q=0$

Finally the quintic equation $ax^5+bx^4+cx^3+dx^2+ex+f=0$ can be reduce to $y^5+py+q=0$, where $p,q\in\Bbb C$.

My question is about, what is the highest value of $n$ be to reduce a general polynomial $a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0=0$ to $y^n+py+q=0\,?$ What must be the highest $n$ for this to be applicable?Is it true that it should be $n\leq 5$?

Some related posts How to transform a general higher degree five or higher equation to normal form? and Is it possible to "depress" any term in a polynomial with a suitable substitution? — Sil, Nov 29 '22 at 22:55
There is the Bring quintic form for any quintic into $x^5+ax+b=0$, but this article implies that not all sextics can be put into $x^6+ax+b=0$ when there would be a closed form for $x$ with a single series hypergeometric function. — Тyma Gaidash, Nov 29 '22 at 23:53
Since Tyma mentioned sextics, then the one-variable quintic $x^5+x+a = 0$ can be solved by the one-variable generalized hypergeometric function, while the two-variable sextic $x^6+x^2+ax+b = 0$ can be solved by the two-variable Kampé de Fériet function. This old post discusses the latter. — Tito Piezas III, Jun 23 '23 at 15:58

What is the highest value of $n$ be to reduce a general polynomial $a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0=0$ to $y^n+py+q=0 ?$

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