An Untrustworthy T-Shirt

Among the many Australian mathematics organisations that are making matters worse rather than better, the Australian Mathematics Trust must not be included. AMT is great, a rare beacon of hope. A beacon somewhat dimmed, it is true, by the fact that the AMT guys have an average age of about 95. Still, any beacon in a storm, or whatever.

Apart from their wonderful work on the Australian Mathematics Competition and their associated endeavours, AMT sells excellent books on problem-solving, as well as some very cool (and some very uncool) mathematical t-shirts. One shirt, however, is particularly eye-catching:

This shirt celebrates Norwegian mathematician Niels Henrik Abel and his 1824 proof of the non-existence of a solution in radicals of the general quintic equation. That is, in contrast to the quadratic formula, and to the cubic and the quartic analogues, there does not exist a quintic formula. It’s a pretty shirt.

It’s also a pretty weird shirt. AMT’s blurb reads

This T Shirt features Abel’s proof that polynomials of order five or higher cannot be solved algebraically.

Stylewise, it is probably a good thing that Abel’s “proof” doesn’t actually appear on the shirt. What is not so good is the sloppy statement of what Abel supposedly proved.

Abel didn’t prove that “[polynomial equations] of order five or higher cannot be solved algebraically”. What he proved was that such equations could not generally be solved, that there’s no general quintic formula. In particular, Abel’s theorem does not automatically rule out any particular equation from being solved in terms of radicals. As a very simple example, the quintic equation

    \[\boldsymbol{x^5-2x=0}\]

is easily shown to have the solutions \boldsymbol{0,\pm\sqrt[4]2,\pm\sqrt[4]2i}.

Which brings us back to AMT’s t-shirt. Why on Earth would one choose to illustrate the general unsolvability of the quintic with a specific equation that is solvable, and very obviously so?

Even good guys can screw up, of course. It’s preferable, however, not to emblazon one’s screw-up on a t-shirt.

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