1 The real subfield and the class numbers
For a number field \(K\), write
\[ K^+ := \operatorname {maximalRealSubfield}(K). \]
In the cyclotomic case this is the fixed field of complex conjugation; for \(K=\mathbb {Q}(\zeta _p)\) it is the usual maximal real subfield.
For a number field \(K\), the notation \(h(K)\) is the cardinality of the ideal class group of \(\mathcal O_K\):
\[ h(K)=\# \mathrm{Cl}(()\mathcal O_K). \]
For a CM field \(K\), the plus class number is
\[ h^+(K)=h(K^+). \]