# Constructability of Roots (esp w/Hilbert Tools)

## Main Question or Discussion Point

ello all,
This is largely a repost of something I posted on r/math, but didn't seem to find any luck there with two of my questions, so I'm asking it again here with the hopes that someone here can answer my questions. Thanks!

I'm studying for a final in geometry, and I know I'm gonna get a question of the form "Let a be a real root of p(x). Is a constructable by ruler and compass? Is a constructable by Hilbert tools?" - Where p(x) is some easy to factor polynomial (judging from past problems, highly likely its a quartic easily solvable by substitution into a quadratic).

Unfortunately, I missed the day of class where we went over this, so I have no notes on it, and it is also pretty much the only topic in the class where we strayed from the textbook, so I have no help there. Also the internet hasn't been the most helpful.

I basically just have three questions.

1) Assuming I can get answers in terms of radicals via the quadratic equation, the (real) roots are always constructable by ruler and compass, right?

2) Furthermore, is it enough to show that the root is not constructable by Hilbert tools by showing it cannot be simplified into root(1+b2 )? (where b is some number)

3) In addition, I seem to remember talking about some additional restraint or conclusion you can reach when you know that not all of the solutions to the polynomial are real, but I cannot remember what that conclusion actually is, can someone remind me of that?

Thanks so much, and if you have more questions I'm happy to answer them.