Proving the Constructibility of Numbers: Hints and Proofs

[b0mb0nika]

Hi,

I was trying to prove the following theorem:

if x is a constructible number <=> it can be obtained from Q by taking a the square root a finite number of times ( or applying a finite # of field operations).

I managed to get the proof for <= this way, but I am not really sure on how to proceed to prove in the other direction =>.

Could anyone give me some hints/ proofs ?

thanks

 

[matt grime]

What is your definition of a constructible number? (Mine is precisely the one above that you're trying to prove is equivalent to yours.)

 

[b0mb0nika]

What is your definition of a constructible number? (Mine is precisely the one above that you're trying to prove is equivalent to yours.)

This is what i thought the definition of a constructible number is :

A real number is constructible if and only if, given a line segment of unit length, one can construct a line segment of length | r | with compass and straightedge.

So then a line segment would be constructible ( by using the thm that i stated before) from Q ( as rational numbers are always constructible) by taking the sqrt a finitely # of times. ..

SO <= IF A=sqrt (a) ( a in Q) its easy to show that you can draw the length sqrt A. And we can extend this to taking the sqrt finitely many times.

I just don't know how to show that if A is constructible then A is egual to sqrt(sqrt(...(a) for some a in Q. ( finitely many sqrt's )

 

[StatusX]

Constructing numbers amounts to looking at the points of intersection of various lines and circles, ie, the solutions of certain pairs of equations. What can you say about these equations?