# Constructible numbers

1. Apr 12, 2007

### 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 im not really sure on how to proceed to prove in the other direction =>.

Could anyone give me some hints/ proofs ?

thanks

2. Apr 12, 2007

### 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.)

3. Apr 12, 2007

### b0mb0nika

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 jsut dont 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 )

4. Apr 12, 2007

### 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?