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Constructible numbers

  1. Apr 12, 2007 #1

    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 ?

  2. jcsd
  3. Apr 12, 2007 #2

    matt grime

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    What is your definition of a constructible number? (Mine is precisely the one above that you're trying to prove is equivalent to yours.)
  4. Apr 12, 2007 #3
    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 )
  5. Apr 12, 2007 #4


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    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?
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