Let \(p\) be a prime number, and let \(K\) be a Galois extension of degree \(p\) of the field of rational numbers \({\mathbb Q}\). Let \(C\) be the ideal class group of \(K\). Let \(\sigma\) be a generator of \(\text{Gal}(K/ \mathbb Q)\), and let \(C^{1- \sigma}=\{a^{1-\sigma}: a\in C\}\). Suppose exactly \(t\) finite primes ramify in \(K/\mathbb Q\), where \(t\) is a positive integer. Then the genus group \(C/C^{1-\sigma}\) is an elementary abelian \(p\)-group with rank equal to \(t-1\) (or possibly \(t-2\) if \(K\) is real quadratic), and \(C^{1-\sigma}\) is called the principal genus. Let \(H=C^{1-\sigma}\), and let \(R\) be the \(p\)-rank of \(H/H^{1-\sigma}.\)
This paper states theorems which indicate how likely it is that \(R=0,1,2,... \). The formulas in these theorems are analogous to conjectured formulas of Cohen and Lenstra for the prime-to-\(p\) part of \(C\), thus suggesting that the Cohen-Lenstra conjectures [H. Cohen and H. W. Lenstra jun., Heuristics on class groups of number fields. Number theory, Proc. Journ. arith., Noodwijkerhout/Neth. 1983, Lect. Notes Math. 1068, 33–62 (1984; Zbl 0558.12002)] might be extended to the principal genus in the ideal class groups of Galois extensions of \(\mathbb Q\) of prime degree \(p\).