Imaginary quadratic field

An imaginary quadratic field is a number field of degree two over $\mathbb Q$ with no embedding into $\mathbb R$ . It has the form

K=\mathbb Q(\sqrt{-d}),\qquad d>0\text{ square-free}.

Here square-free fixes a unique positive integer parameter for the isomorphism class.

Elements and embeddings

Each element is uniquely $a+b\sqrt{-d}$ , with $a,b\in\mathbb Q$ . Its two complex embeddings send $\sqrt{-d}$ to $i\sqrt d$ and $-i\sqrt d$ ; they are exchanged by complex conjugation.

Example

For $d=1$ , the field is $\mathbb Q(i)$ . Its ring of integers is a smaller subset than the field: denominators are generally not allowed.

References

J. S. Milne, Algebraic Number Theory, v3.08. Author’s text, Introduction, quadratic-field examples.