Abel

NIELS HENRIK ABEL (1802 to 1829)

A polynomial in x of degree n is an expression of the form

where a_i, i=0,1,2,...,n (a_n≠ 0), are given real numbers and known as the coefficients.

A polynomial equation p(x)=0 can occur frequently in real life and a common task is to find solutions, i.e. values of x which satisfy the equation.

A polynomial equation of degree n will have n solutions, or roots. However some of these may be complex rather than real. If any are complex they will occur as conjugate pairs, i.e. numbers of the form a±bi where a and b are real and i²=-1.

A polynomial equation of degree 1, i.e. of the form ax+b=0, (a≠0), where a and b are real coefficients, is called linear and has the obvious solution x=-b/a.

A polynomial equation of degree 2, say ax^2+bx+c=0 (a≠0) is called quadratic, and since ancient times has been known to have the solution

(which can easily be proved if you know a technique known as completing the square). Note that if b²<4ac the expression under the square root sign is negative and this yields a pair of complex conjugates (this corresponds graphically to a parabola with a maximum value, and the maximum is negative, or a parabola which has a minimum which is positive). If b²=4ac there are two equal real roots and this corresponds graphically to a case where the quadratic has a maximum or minimum touching the x-axis.

Polynomial equations of degrees 3, 4 and 5 are known as cubics, quartics and quintics respectively. In the 16th century, Italian mathematicians including Tartaglia, Ferrari and Cardano had found similar formulae for cubic and quartic equations (solutions in terms of their coefficients) but for centuries mathematicians tried in vain to find a formula for the solution of a quintic.

Finally, in 1824, Abel, the great Norwegian mathematician who unfortunately died at a very young age, proved that it is impossible to find such a formula for a quintic, such as the one on the T Shirt, or polynomial equations of higher degree. Incidentally, since complex solutions only appear in (conjugate) pairs, polynomial equations when the degree is odd must have at least one real root. It is easy for you to find, by substituting small whole numbers, a real solution of the quintic on the T Shirt, but you won't find any other real solutions in this case as the other solutions of this quintic comprise two complex conjugate pairs.

In his short life Abel made other significant discoveries, including a proof of the binomial theorem for all real numbers, not just for rational numbers, as had been proved by Euler.

Efforts to recognise the achievements of Abel in his own country, Norway, by naming a major prize after him, began in the decades following his death in 1829, but were not successful until the Abel Prize for mathematics was established by the Norwegian Government in 2001. They hope it will achieve the same status as the Nobel Prize. Norway has also produced postage stamps and a coin commemorating Abel.

Peter Taylor

March 2012