Direct proof that π\pi is not constructible

Is there a direct proof that π is not constructible, that is, that squaring the circle cannot be done by rule and compass?

Of course, π is not constructible because it is transcendental and so is not a root of any polynomial with rational coefficients. But is there a simple direct proof that π is not a root of polynomial of degree 2n with rational coefficients?

The kind of proof I seek is one by induction on the height of a tower of quadratic extensions, one that ultimately relies on a proof that π is not rational. Does any one know of a proof along these lines or any other direct proof?

I just want a direct proof that π is not constructible without appealing to transcendence.


I’ve just read in the book The Number π by Eymard and Lafon that no such proof is known.

“The proof that it is impossible to square the circle does not involve direct demonstration of the non-constructibility of the number π. As far as we are aware, it is not known how to do this! One proves that it is not algebraic, which is much more restrictive, then one uses the fact that a constructible number is algebraic.” [§4.2, p. 134]

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