Constructability of Roots (esp w/Hilbert Tools)

[C. Caesar]

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 going to 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.

 

[jvicens]

Hello there,

I would be happy to help answer your questions about constructability in geometry. Firstly, to answer your first question, yes, any real root of a polynomial that can be solved using the quadratic formula can be constructed using ruler and compass. This is because the quadratic formula gives us the exact values of the roots, which can then be constructed using ruler and compass.

For your second question, it is not enough to show that the root cannot be simplified into root(1+b^2) to prove that it is not constructable by Hilbert tools. This is because there are other methods of constructing numbers using Hilbert tools, such as using the Pythagorean theorem or constructing the geometric mean of two numbers. So, to prove that a root is not constructable by Hilbert tools, you would need to show that it cannot be constructed using any known method.

Lastly, for your third question, if not all of the solutions to a polynomial are real, then we can conclude that the polynomial has complex roots. This means that the roots cannot be constructed using ruler and compass, as ruler and compass constructions are limited to real numbers. This is because complex numbers involve the use of imaginary numbers, which cannot be constructed using ruler and compass.

I hope this helps answer your questions. If you have any further questions, please feel free to ask and I will do my best to assist you. Good luck on your final!