An elementary proof of the theorem on the imaginary quadratic fields with class number 1 (2401.07192v1)

Abstract: Let $D$ be a square-free integer other than 1. Let $K$ be the quadratic field ${\mathbb Q}(\sqrt D)$. Let $\delta \in {1,2}$ with $\delta=2$ if $D\equiv 1 \pmod 4$. To each prime ideal $\mathcal P$ in $K$ that splits in $K/\mathbb Q$ we associate a binary quadratic form $f_{\mathcal P}$ and show that when $K$ is imaginary then $\mathcal P$ is principal if and only if $f_{\mathcal P}$ represents $\delta^2$, and when $K$ is real then $\mathcal P$ is principal if and only if $f_{\mathcal P}$ represents $\pm \delta^2$. As an application of this result we obtain an elementary proof of the well-known theorem on the imaginary quadratic fields with class number 1. The proof reveals some new information regarding necessary conditions for an imaginary quadratic field to have class number 1 when $D\equiv 1 \pmod 4$.