An elementary proof of the theorem on the imaginary quadratic fields with class number 1

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 $D\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}4)$. To each prime ideal $\mathcal{P}$ in $K$ that splits in $K\mathrm{/}\mathbb{Q}$ 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 $D\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}4)$.