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Heck, it's got ''three'' cube roots.
:The view that "all proofs are equivalent no matter what theories they use" may be appealing to some, but it does not withstand scrutiny. Proofs relying on fewer assumptions are obviously preferable.--[[User:Aschlafly|Andy Schlafly]] 22:44, 26 August 2009 (EDT)
:The view that "all proofs are equivalent no matter what theories they use" may be appealing to some, but it does not withstand scrutiny. Proofs relying on fewer assumptions are obviously preferable.--[[User:Aschlafly|Andy Schlafly]] 22:44, 26 August 2009 (EDT)
The notion that complex analysis depends on a purported uniqueness of the square root of -1 needs to be put to rest. There is no assumption that the square root of -1 has to be unique. And a good thing, too, because it isn't unique. No one denies that -1, like all nonzero numbers, has two square roots, that are negatives of each other. All nonzero numbers also have three cube roots, related by de Moivre's formula, and so on. Complex analysis does not rely on any such uniqueness. The complex numbers (or complex plane, or Argand diagram) is constructed from a special formal element "i", which simply serves as the second basis element of the vector space being constructed. The "square root of -1" and "i" are not the same thing. There are two square roots of -1, but only one "i". The square root of -1 doesn't even exist until the complex plane has been constructed, so it couldn't have been used in the construction.
The complex numbers, so constructed, are every bit as rigorous as the rational numbers, the real numbers, Hilbert spaces, Frechet spaces, Lie groups, symplectic groups, etc.
If there were any ambiguity in the interpretation of the "imaginary part" of a complex number, many fields of mathematics, physics, and engineering would fall apart. To pick just a few examples, the field of linear circuits in electrical engineering depends on the Laplace transform, which would not be well-defined if the imaginary unit "i" were not well-defined. Also, there could be no agreement on what the Schordinger equation of quantum mechanics means, or how the various quantum mechanical operators work. The Fourier transform would not be well-defined.
There may well be reasons why one would prefer a theorem of number theory that does not involve complex analysis, but such a proof's lack of rigor is not one of them.
We need to work out a careful survey of the things that make some proofs more complicated than other proofs of the same theorem, and how the word "elementary" is used to describe this, and what the various ramifications of this usage are. I'm willing to work on this. [[User:PatrickD|PatrickD]] 23:06, 26 August 2009 (EDT)