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ON CLASS NUMBER ONE FOR THE REAL QUADRATIC FIELD $\mathbb{Q}\left(\sqrt{a^2+p}\right)$
Keywords:
class number, real quadratic fieldDOI:
https://doi.org/10.17654/0972555522021Abstract
Let $d$ be a prime with $d \equiv 1(\bmod 4), K=\mathbb{Q}(\sqrt{d})$ and $\mathcal{O}=\mathbb{Z}[\sqrt{d}]$ which is an order in the ring $\mathcal{O}_K$ of integers of $K$. Using the bijection between matrix conjugations over $\mathbb{Z}$ with characteristic polynomial $f(x)=x^2-d$ and $\mathcal{O}$-ideal classes, we study the relation between $\mathcal{O}$-ideal class group and $\mathcal{O}_K$-ideal class group. Assume that $d$ satisfies the condition: if $d=a^2+b c$ with $a$ even, then $b c=1$ or $|b c|=p$ is an odd prime. Then we prove that the class number of $K$ is one.
Received: April 5, 2022
Accepted: May 19, 2022
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