This page intentionally left blank.

On August 5, 2002, the world celebrated the bicentenary of the birth of one of history’s most eminent mathematicians, the Norwegian Niels Henrik Abel (1802–1829). He died, barely 26 years old, of tuberculosis. Brief though his life was, Abel’s work was extremely fruitful. An important encyclopedia for mathematics mentions the name Abel and the adjective “abelian” close to 2,000 times.

The works of Abel are so significant that in 2001 Thorvald Stoltenberg, then Norwegian prime minister, announced the establishment of the Abel Endowment Fund, which would henceforth award an annual prize of 800,000 euros in his name. This prize, which models itself on the Nobel Prize, was meant to become the most significant award in the field of mathematics.

Abel grew up in Gjerstad, a town in the southern part of Norway, as the second oldest in a family of seven children. His father was a Lutheran priest who at one time also acted as a member of the Norwegian parliament. Until the age of 13, Niels was educated at home by his father. It was only when the teenager started to attend a church school in Christiana, 120 miles away, did his talents really became apparent. A mathematics teacher recognized the unusual gifts of the young boy and encouraged him as best he could.

When Abel was 18 years old, his father died and all of a sudden Abel found himself forced to support his family, which he did by offering private lessons in basic mathematics and performing odd jobs. Thanks to the financial help of his teachers, Abel was able, in 1821, to enroll at the University of Christiana, which was later to become the University of Oslo. It did not take long before Abel started to outshine his teachers. His first major achievement, however, proved to be in error. Abel believed that

he had found a method to solve equations of fifth degree and submitted his paper for publication to a scientific journal. The editor could not understand the solution, however, and asked Abel for a numerical example.

Abel set out to meet this request but soon became aware of a mistake in the derivation. The error proved to be of some benefit, though. While trying to make the correction, Abel realized that it is simply impossible to solve an equation of fifth or higher degree by means of a formula. To arrive at, and prove, this conclusion Abel made use of a concept called group theory, which would later develop into a very important branch of modern mathematics.

Abel published this paper at his own expense. He then undertook a journey to Germany with the financial support of the Norwegian government in order to call on the famous mathematician Carl Friedrich Gauss in Göttingen. Gauss, however, never read the paper that Abel had sent him ahead of his visit. Furthermore, he let him know in no uncertain terms that he had no interest whatsoever in a meeting. Disappointed, Abel continued his journey on to France, a side trip that had a fortuitous side effect. En route to Paris he made the acquaintance of the engineer August Leopold Crelle in Berlin, who was to become a very close friend and supporter. The Journal für Reine and Angewandte Mathematik (Journal for Pure and Applied Mathematics), which was founded by Crelle and continues to appear to this very day, published many of Abel’s highly original papers.

The French colleagues whom Abel tried to look up proved to be no more hospitable than the German professor. A work on elliptic functions that Abel had produced and sent by way of introduction to Augustin Cauchy, the leading French mathematician of the day, was not even noticed. His paper fell into oblivion and was finally lost altogether. Despite this disappointment, Abel persisted, stayed on in Paris, and did everything he could to gain recognition for his work. He was desperately poor and could only afford a single meal a day.

But in the end none of his sacrifices paid off. Crelle

tried everything to convince his friend to settle in Germany, but Abel, sick and without a penny, returned to his home country. After Abel’s departure, Crelle set about to find an academic position for his friend, and at long last his efforts bore fruit. In a letter dated April 8, 1804, he was overjoyed to be able to inform his friend that the University of Berlin was offering him a teaching position as a professor. Unfortunately, it was too late: Niels Henrik Abel had died of tuberculosis two days earlier.

From the many concepts connected to Abel, let us briefly mention the concept of the “abelian group.” Modern algebra defines a set of elements as a group if these elements can be connected to each other with the help of an operation. Four conditions must be fulfilled: First, the result of the operation must also be an element of the group. Second, the operation must be “associative,” which means that the order in which two successive operations are performed does not matter. Third, a so-called neutral element has to exist, which leaves the result of the operation unchanged. Fourth, each element must have an inverse. For example, the whole numbers form a group under the operation of addition. Here are the reasons why: The sum of two whole numbers is also a whole number; the operation is associative since (a + b) + c = a + (b + c); the number zero is the neutral element because a number plus zero leaves the number unchanged; and the inverse element of, say, 5 is −5. Rational numbers (whole numbers and fractions) do not form a group under multiplication, even though the product of two rational numbers is also a rational number (for example, 2/3 times 3/7 equals 6/21), the inverse element of 5 is 1/5, and the neutral element, in this case, is 1. That is because 0 has no inverse.

Groups can be subdivided into “abelian” and “nonabelian” groups. A group is called abelian if the elements, when connected to one another, can be interchanged (for example: 5 + 7 = 7 + 5). An example of elements that form a “nonabelian” group are the rotations of dice. If one rotates a die around two different axes in sequence, it certainly matters in which order these rotations take place.

Try it and see for yourself. Take two dice and place them on the table in identical positions. Rotate the first die around the vertical and then around a horizontal axis. Then rotate the second die in the same directions, but first around the horizontal axis and then around the vertical. You will note that the faces of the dice point in different directions. Hence the group of rotations of dice are ”nonabelian.” It is this particular fact, among others, that makes the solution of Rubik’s famous cube so devilishly tricky!