= Quadratic integer
A <quadratic integer> is an <algebraic integer> in a <quadratic number field> $K=\mathbb Q(\sqrt d)$, where $d$ is a squarefree <integer> other than $0,1$. Its <ring of integers> is $\mathbb Z[(1+\sqrt d)/2]$ if $d\equiv1\pmod4$, and $\mathbb Z[\sqrt d]$ otherwise. Thus $(1+\sqrt5)/2$ is a <quadratic integer>, since it satisfies the monic equation $X^2-X-1=0$.
For $\alpha=a+b\sqrt d$, its conjugate is $a-b\sqrt d$. Its <field trace> and <field norm> are $2a$ and $a^2-db^2$; their integrality characterizes the <algebraic integers> in this degree-two extension. These formulas connect <Diophantine equations> to arithmetic in a larger <ring>.
Back to article page
