Quadratic field

Remarks

Every quadratic field is $\Q(\sqrt{d}) = \set{a + b\sqrt{d} : a, b \in \Q}$ for exactly one square-free integer $d$ with $d \ne 0, 1$. It is called real if $d > 0$ and imaginary if $d < 0$. Example: $\Q(\sqrt{2})$; the element $\sqrt{2}$ is a root of the irreducible polynomial $x^2 - 2$, so $1, \sqrt{2}$ is a basis and the degree is $2$. Non-example: $\Q(\sqrt[3]{2})$ has degree $3$, because $x^3 - 2$ is irreducible over $\Q$; so it is a number field but not a quadratic field. The ring of integers of $\Q(\sqrt{d})$ is $\Z[(1 + \sqrt{d})/2]$ if $d \equiv 1 \pmod 4$, and $\Z[\sqrt{d}]$ if $d \equiv 2$ or $3 \pmod 4$. The discriminant is $d$ in the first case and $4d$ in the second.