6 Algebraic Number Fields

We can look at the factorisation problem as the study of the group of non-zero rationals; writing every element in terms of the generators (the prime numbers and -1) and taking into account the relation (- 1)² = 1. The study of the unit group in $\mathbb {Z}$/N$\mathbb {Z}$ can be identified with the study of a suitable quotient of a suitable subgroup (elements prime to N) of this group. We now ask how this group can be generalised. One natural idea is to use algebraic number fields. An algebraic number is an ``object'' (we will be more specific later) that satisfies a polynomial equation with rational (equivalently integer) coefficients (we should actually insist on irreducibility of the equation). We can represent such objects as we will see below. However, it turns out that studying groups of algebraic numbers is not quite the same as studying the generalised factorisation problem; that involves the study of divisors or ideals and their groups.