An example quintic whose roots cannot be expressed by radicals is $x^5 - x + 1 = 0$.
I asked a geometry question about a fifth degree equation long time ago . I had an equation in the question. It is $\sin(5\beta)+\sin(\beta)=1$ and It can be expressed as a fifth degree equation $16P^5-20P^3+6P=1$. It was solved by radicals in my previous question.
Is it possible to create a fifth degree equation by using circles and lines that cannot be solved by radicals such as $x^5-x+1=0$ ?
In other words;
If an fifth degree equation cannot be solved by radicals, Can we express the roots of the equation (at least one real root) by drawing circles , lines in any way? ( In my example, I used angles to create a fifth degree equation)?
Thanks for answers