quadratic field

The simplest number fields beyond the rationals themselves are the ones you get by adjoining a single square root, like sqrt(2) or sqrt(-1). These quadratic fields are the laboratory of algebraic number theory: small enough to compute everything by hand, rich enough to display ramification, class groups, units and reciprocity in their cleanest form.

A quadratic field is a number field K of degree 2 over Q. Every such field has the form Q(sqrt(d)) for a unique squarefree integer d not equal to 1; it is real (with two real embeddings) when d > 0 and imaginary (with one pair of complex embeddings) when d < 0. Its ring of integers is Z[sqrt(d)] when d ≡ 2 or 3 (mod 4) and the larger ring Z[(1 + sqrt(d))/2] when d ≡ 1 (mod 4).

Quadratic fields are exactly the abelian extensions of Q with Galois group Z/2Z, and their arithmetic is governed by the quadratic reciprocity law: how a prime splits in Q(sqrt(d)) is determined by a Legendre symbol, hence by congruences modulo the discriminant. Class numbers of imaginary quadratic fields are among the deepest objects in the subject.