## Niels Abel (1802 - 1827)

Niels Abel began his career in mathematics reading the works of Newton, Euler and Gauss. He went to University on a scholarship, where he started working on quintic equations, aged only 19.

Sadly, Abel contracted tuberculosis while still a young man and died in 1829 aged only 26. However, he continued to contribute to maths after his death, when in 1830, an unpublished paper of his on the algebraic solution of equations was discovered.

It described in essence the same theory that Galois had outlined earlier that year, and therefore showed that Abel had pre-empted Galois by some years. Abel and Jacobi were subsequently awarded the Grand Prix by the Paris Academy for their work.

## Abel’s mathematics

Abel is one of that small band of mathematicians that have given their name to an area of mathematics. Euclid - as in Euclidean geometry - is another. It is a rare honour. Abel’s honour is concerned with groups: an abelian group is one in which the order of the elements of the group is unimportant.

The equation 3x+7=13 has degree 1, since it involves only powers of 1 in the unknown x;

The equation 2a^2-3a+7=0 is, bizarrely, called a quadratic - the highest power of the unknown a is 2;

A quintic equation has degree 5:

The equation z^5-4z^2+3z=7 is called a quintic equation since the highest power of the unknown z that occurs is 5.

Equations in which the following highest powers occur, have corresponding names: 3 cubic, 4 quartic, and so on.

He proved that unlike quadratic, cubic, and quartic equations, the general quintic cannot be solved algebraically in terms of a finite number of additions, subtractions, multiplications, divisions, and root extractions.

Abel’s work on quintic equations is fundamental to the subjects that are included within abstract algebra.

Not content with quintic equations, Abel also explored, and contributed greatly to, other areas of maths, including:

• the theory of elliptic functions and elliptic integrals (which he worked on with Jacobi);
• integrals;
• infinite series.