Quadratic field

A quadratic number field is a number field of degree of a field extension two, necessarily of the form $\mathbb{Q}(\sqrt{d})$ for a square-free integer $d\ne 0,1$. Its two field embeddings send $a+b\sqrt{d}$ to $a\pm b\sqrt{d}$. If $d>0$, both images lie in the real numbers, giving a totally real number field; if $d<0$, they form a pair related by complex conjugation, giving an imaginary quadratic field. The ring of integers of a quadratic field depends on $d$ modulo four.^[ref]

The ring of integers is $\mathbb{Z}[\sqrt{10}]$, its class number is two, and its units are $\pm (3+\sqrt{10}{)}^n$ for $n\in \mathbb{Z}$. Factoring a norm-six principal ideal above 2 and 3 gives all solutionsRestriction to nonnegative $n$ gives only the positive-coordinate branch up to simultaneous sign. For example, $4-\sqrt{10}$ is a norm-six element requiring $n=-1$; adding conjugates repairs a nonnegative-index presentation.

An imaginary quadratic field is a quadratic number field with no real embedding. Its two complex embeddings are interchanged by complex conjugation. The endomorphism ring of an elliptic curve in characteristic zero, when larger than the integers, is an order in such a field; this is complex multiplication.

A fundamental discriminant is an integer that occurs as the field discriminant of a quadratic number field. For square-free $d$, it is $d$ when $d\equiv 1(\mathrm{mod}4)$ and $4d$ otherwise.