> Dear Colleagues
>
> Everyone knows that one of the successes of modern
> algebra has been to prove the impossibility of the classical
> problems of doubling the cube and trisecting the angle with
> ruler and compass. The key point is that if a is
> constructible, then a is a root of an irreducible polynomial
> f(x) with rational coefficients and degree a power of 2.
Actually a stronger statement holds: a is contained in a galois extension
of the rationals with the Galois group a 2-group (and conversely).
But of course the fact that the degree is a power of 2 suffices for proving
the impossibility of trisection and duplication.
> In modern algebra texts this is a simple exercise using the
> degree of a finite field extension.
>
> Credit for the first proof of this result is usually given to
> Wantzel for his paper in the Liouville Journal of 1837. So I
> thought I would read his paper, and I found an error in his
> proof. He supposes that a can be obtained by solving a
> sequence of n quadratic equations, and then shows that a
> is a root of a polynomial f(x) of degree 2^n . Then,
> assuming that his chain of quadratic equations is minimal,
> in the sense that one cannot omit any one of them to get a,
> he claims f(x) is irreducible. He shows correctly that if
> g(x) is any other polynomial having a as a root, then
> every root of f(x) is also a root of g(x). However, this
> does not imply f(x) irreducible (as he wants) because f(x)
> could have multiple roots. A simple example is given by
>
> a = sqrt(3 + sqrt(2)) - sqrt(3 - sqrt(2)) .
>
> In this case we obtain a by a tower of three quadratic
> equations, but the minimal polynomial of a has degree 4.
>
But if we choose g(x) above to be of minimal degree, it follows easily
that f(x) is a power of g(x), so that the degree of g(x) is also a power
of 2, and this suffices. It is possible that Wantzel's contemporaries
noticed the gap, and also that it could be easily corrected, so they
still credited him with the proof. Such gaps, which the reader has to,
and usually can, fill, occur often.
Avinoam Mann
> So my question is, has this been noticed before? And since
> Wantzel's proof has an error, to whom do we credit the
> first correct proof?
>
> The earliest correct proof I have found is Julius Petersen in
> his Theorie der algebraischen Gleichungen, 1878, using
> results of his thesis 1871 (which I have not been able to
> consult directly). Klein's proof in his Vortrage 1895 is
> hardly more than an expanded elaboration of Petersen's
> proof. Pierpont 1895 gives an independent proof,
> apparently unaware of either Wantzel or Petersen.
>
> The problem in Wantzel's paper also raises some broader
> questions: what constitutes sufficient proof for a result in
> any historical period, and what are the different ways in
> which a proof of one period may be considered insufficient
> by a later generation of mathematicians? Was the error in
> Wantzel's paper not noticed by all those who give him
> credit, or was it knowingly tolerated and accepted as a
> correct proof?
>
> As usual, any comments or feedback will be welcome.
>
> Robin Hartshorne
> Berkeley
>
>