Quadratic Integer - Quadratic Integer Rings

Quadratic Integer Rings

Fixing a square-free integer D, the quadratic integer ring Z = { a + ω b : a, b ∈ Z}  is a subring of the quadratic field . Moreover, Z is the integral closure of Z in . In other words, it is the ring of integers of Q(√D) and thus a Dedekind domain. The quadratic integer rings usually form the first class of examples on which one can build theories, inaccessible in the general case, for example the Kronecker–Weber theorem in class field theory, see below.