The Gaussian integers showed the power of enlarging the integers with a new number, i. Algebraic number theory makes this systematic: adjoin a root of any polynomial and study the arithmetic of the resulting world. The natural homes for this are number fields.

Algebraic numbers

A number is algebraic if it is a root of a polynomial with rational coefficients. Familiar examples include \sqrt{2} (root of x^2 - 2), i (root of x^2 + 1), and the golden ratio. Numbers that are not algebraic — like \pi and e — are called transcendental. Almost all real numbers are transcendental, yet algebraic numbers are the ones number theory can grasp.

What a number field is

A number field is what you get by adjoining an algebraic number \alpha to the rationals — the smallest field containing both \mathbb{Q} and \alpha, written \mathbb{Q}(\alpha). The simplest beyond \mathbb{Q} are the quadratic fields:

Each is a finite-dimensional vector space over \mathbb{Q} — here 2-dimensional, with basis \{1, \sqrt d\}. That finiteness (the degree of the field) is what keeps everything under control.

Why generalise this far

Many integer problems become transparent in the right number field. Sums of two squares live naturally in \mathbb{Q}(i); Pell's equation lives in \mathbb{Q}(\sqrt d); Fermat's equation was attacked in fields built from roots of unity. But to do arithmetic we first need the right notion of "integer" inside a number field — the algebraic integers, next.