Quadratic Rings

From an abstract algebraic perspective, an explanation for this can be given as follows: since √ D is irrational, the polynomial t −D is irreducible over Q. Since the ring Q[t] is a PID, the irreducible element t − D generates a maximal ideal (t−D), so that the quotient Q[t]/(t−D) is a field. Moreover, the map Q[ √ D]→ Q[t]/(t −D) which is the identity on Q and sends √ D 7→ t is an isomorphism of rings, so Q[ √ D] is also a field. We may write Q[ √ D] = {a + b √ D | a, b ∈ Q}, so that a basis for Q[ √ D] as a Q-vector space is 1, √ D. In particular Q[ √ D] is two-dimensional as a Q-vector space: we accordingly say it is a quadratic field.