"He took up the problem of the division of the lemniscate (solving xn - 1 = 0 is the equivalent of the problem of the division of the circle into n equal arcs) and arrived at a class of algebraic equations... Abelian equations, that are solvable by radicals. The cyclotomic equation [xp - 1 = 0, where p is a prime] is an example... In this last work he introduced two notions (though not the terminology), field and polynomial irreducible in a given field. By a field of numbers he, like Galois later, meant a collection of numbers such that the sum, difference, product, and quotient of any two numbers in the collection (except division by 0) are also in the collection. ...A polynomial is said to be reducible in a field (usually the field to which its coefficients belong) if it can be expressed as a product of two polynomials of lower degrees and with coefficients in the field. Abel then tackled the problem of characterizing all equations which are solvable by radicals and had communicated some results... just before death overtook him in 1829."