Quadratic number field Q(sqrt(d))

For a squarefree integer $d\ne0,1$ , the quadratic number field $\mathbb Q(\sqrt d)$ is the quotient field extension $\mathbb Q[x]/(x^2-d)$ . Its elements have unique expressions $a+b\sqrt d$ with $a,b\in\mathbb Q$ .

Real and imaginary cases

For $d>0$ the field has two real embeddings; for $d<0$ it has a pair of conjugate complex embeddings and no real embedding. In both cases the nonidentity rational-field automorphism negates $\sqrt d$ .

Catalogue scope

This is a parameterized family, not a single chosen field. A specific value of $d$ and an embedding must be supplied when a relationship depends on that choice. The case $d=1$ is excluded because the quotient by $x^2-1$ is not a field.