All known proofs that the circle cannot be squared are based on Lindemann's theorem that $\pi$ is not analgebraic number. But this seems to be a case of using an atomic bomb to kill a fly.

What really has to be proved is that $\pi$ is not the root of an algebraic equation with rational coefficients whose degree is a power of $2$. Lindemann proves it for all degrees, which is much much more than what is needed for the classical construction problem.

It is not too hard to prove that $\pi$ is not the root of a quadratic equation, and perhaps some clever induction argument could carry the day. The point is that a direct proof for degree $2^n$ has not only never been published, it does not even seem to have been noticed that this much less powerful result is all that is needed.

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  • Well, sometimes it's needed an atomic bomb to kill a particularly pesky fly. I'm not aware of any "direct" proof that $\;\pi\;$ isn't algebraic. – Timbuc Nov 06 '14 at 13:58
  • True, Timbuc, but it is not clear that one needs an atomic bomb, here. If it is necessary, why is that? Why has nobody tried the simpler case of degree 2^n? – mark Nov 06 '14 at 14:00
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    Is it simpler? How does the "not too hard" proof that $\pi$ isn't of degree $2$ look? – Daniel Fischer Nov 06 '14 at 14:03
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    There is a proof that pi is not the root of a cubic in "Monographs on Modern Mathematics"...the case of the quadratic is even easier... – mark Nov 06 '14 at 14:05
  • @mark, perhaps proving that is as simple as proving the pretty simple-looking statement that in the integers, $\;a^n+b^n=c^n\;$ is impossible if $\;3\le n\in\Bbb N\;$ . – Timbuc Nov 06 '14 at 14:05
  • See the comment of Gerry [here](http://math.stackexchange.com/questions/103786/direct-proof-that-pi-is-not-constructible). – Dietrich Burde Nov 06 '14 at 14:06
  • Granted, timbuc, but WHY?...As a teacher of mine said many years ago in class "It's obvious...it may not be CLEAR that it's obvious, but it's obvious..." – mark Nov 06 '14 at 14:07
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    Thank you, Dietrich...yes, that is precisely my question..very interesting! – mark Nov 06 '14 at 14:10
  • Well, why was FLT so hard to prove being a statement that any clever junior h.s. kid can understand? Simple: I don't have the faintest idea. Of course, I *now* know what kind of heavy weaponry was used for that, but **why** it turned out to be so, I can't say at all. Maybe that small margin made things crazy hard...fortunately enough, otherwise several wonderful branches of mathematics wouldn't have developed or even existed. – Timbuc Nov 06 '14 at 14:12

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