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; feeding it to theorem 3.6 converts \(p\mid h(K)\) into \(p\mid h^-(K)\). The minus class-number criterion (theorem 3.4) rewrites the latter as the existence of a Bernoulli numerator in Kummer’s range divisible by \(p\).