Question about Gauss composition law over PID.

Question

Let $m$ be a square free integer, $\mathbb{Q}(\sqrt{m})$ a quadratic field extension of $\mathbb{Q}$, $\Delta$ is its discriminant and $O_{\mathbb{Q}(\sqrt{m})/\mathbb{Q}}$ its ring of integers. We have the following results (see Lenstra, On the calculation of regulators and class numbers of quadratic fields):

1) Every ideal of $O_{\mathbb{Q}(\sqrt{m})/\mathbb{Q}}$ is of the form $d\times (\mathbb{Z}a+\mathbb{Z}\frac{b+\sqrt{\Delta}}{2})$ with $d, a, b\in\mathbb{Z}$ and $c=\frac{b^{2}-\Delta}{4a}\in\mathbb{Z}$.

2) Let $\displaystyle{I_{1}=\mathbb{Z}a_{1}+\mathbb{Z}\frac{b_{1}+\sqrt{\Delta}}{2}}$ and $\displaystyle{I_{2}=\mathbb{Z}a_{2}+\mathbb{Z}\frac{b_{2}+\sqrt{\Delta}}{2}}$ be two ideals of $O_{\mathbb{Q}(\sqrt{m})/\mathbb{Q}}$, $\displaystyle{c_{1}=\frac{b_{1}^{2}-\Delta}{4a_{1}}}$ and $\displaystyle{c_{2}=\frac{b_{2}^{2}-\Delta}{4a_{2}}}$. Then the ideal $I_{3}=I_{1}\times I_{2}$ is of the following form

$$ I_{3}=d_{3}\times(\mathbb{Z}a_{3}+\mathbb{Z}\frac{b_{3}+\sqrt{\Delta}}{2}) $$

with $\displaystyle{d_{3}=gcd(a_{1},a_{2},\frac{b_{1}+b_{2}}{2})}$, $\displaystyle{a_{3}=\frac{a_{1}a_{2}}{d_{3}^{2}}}$, $\displaystyle{b_{3}=\frac{\alpha a_{1}b_{2}+\beta a_{2}b_{1}+\delta \frac{b_{1}b_{2}+\Delta}{2}}{d_{3}}}$ and $d_{3}=\alpha a_{1}+\beta a_{2}+\delta \frac{b_{1}+b_{2}}{2}; \alpha, \beta, \delta\in\mathbb{Z}$.

These results are related to classical Gauss composition law. My question is if these results remain true when we replace $\mathbb{Q}(\sqrt{m})/\mathbb{Q}$ by any quadratic extension of number fields $\mathbb{L}/\mathbb{K}$ such that $\mathbb{K}$ is of the class number one? and where can I find the proofs if they exist.

Thank you.

Answer 1

Composition in PIDs $R$ is no problem: all you have to do is replace the ring of integers in your favorite proof by $R$. Speiser wrote his thesis on the theory of binary quadratic forms in number fields; the main problem, of course, is generalizing the theory of reduction.

Actually composition of binary quadratic forms works in almost arbitrary domains; the most general version was given by Kneser; see also Schwermer's exposition in The Shaping of Arithmetic after C. F. Gauss's Disquisitiones arithmeticae. Another nice article on composition perhaps closer to your interest is by J. Towber, Composition of oriented binary quadratic form-classes over commutative rings, Advances in Math. 36 (1980), 1–107 .