{ d ∤ d_{E} \{ h(K_{gen}) = \frac{h(K)}{[K_{gen}:K]} \{ \begin{matrix}h(K) = 1 \\ h(K) > 1 \end{matrix} h(K_{gen}) > \frac{h(K)}{[K_{gen}:K]}  \begin{matrix}⋯ & K_{ur} = K \\ ⋯ & K_{ur} = K_{1} \\ ⋯ & K_{ur} = K_{2} \end{matrix} \}K_{ur} = (K_{gen})_{1}

d | d_{E} \{ d = d_{E} d = d′d_{E} (d′ fundamental quadratic discriminant) \begin{matrix}⋯ & K_{ur} = K_{2} \\ ⋯ & K_{ur} = K_{3} \end{matrix} \} K_{ur} ⫌ (K_{gen})_{1}