For a number field \(K\), let \(K^+\) denote its maximal real subfield, the largest subfield of \(K\) all of whose complex embeddings are real. When \(K\) is a CM field, \(K^+\) is the fixed field of complex conjugation, and \([K : K^+] = 2\). For \(K = \mathbb {Q}(\zeta _p)\) this is the usual maximal real subfield \(\mathbb {Q}(\zeta _p + \zeta _p^{-1})\).

For a number field \(K\), the class number \(h(K)\) is the cardinality of the ideal class group of its ring of integers:

For a CM field \(K\), the plus class number is the class number of the maximal real subfield: