# Math Help - An example of 2 quadratic integers in the same quadratic field

1. ## An example of 2 quadratic integers in the same quadratic field

What's an example of two quadratic integers in the same quadratic field for which N(a)|N(b), yet a does not divide b?

2. ## Re: An example of 2 quadratic integers in the same quadratic field

Fix a squarefree integer $D$. Then $\mathbb{Q}(\sqrt{D})$ is a quadratic field. So, $a,b\in \mathbb{Z}(\sqrt{D})\subseteq \mathbb{Q}(\sqrt{D})$. With $N(a)|N(b)$ gives a relationship between them. Let $\alpha_1,\alpha_2,\beta_1,\beta_2 \in \mathbb{Z}$ such that $a = \alpha_1 + \beta_1\sqrt{D}, b = \alpha_2 + \beta_2\sqrt{D}$. Then $N(a) = \alpha_1^2 - D\beta_1^2, N(b) = \alpha_2^2 - D\beta_2^2$. There exists $k \in \mathbb{Z}$ such that $\alpha_2^2-D\beta_2^2 = (\alpha_1^2-D\beta_1^2)k$. If $a$ does not divide $b$, then there exists a unique $q,r\in \mathbb{Z}(\sqrt{D})$ with $b=aq+r$ and $0 (note that $N(r) \neq 0$, as this would imply divisibility). Use these relationships to find an example.