2 The total class-number criterion
For an odd prime \(p\) and the \(p\)th cyclotomic field \(K\),
\[ p\mid h(K) \quad \Longleftrightarrow \quad \exists k,\ 1\le k,\ 2k\le p-3,\quad p\mid \operatorname {num}(B_{2k}). \]
Proof
The implication \(p\mid h^+(K)\Rightarrow p\mid h^-(K)\) is theorem 1.3. Therefore theorem 3.6 rewrites \(p\mid h(K)\) as \(p\mid h^-(K)\). The result follows from the minus class-number criterion theorem 3.4.