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. 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.
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