Class Groups of Dihedral Extensions

Let L/F be a dihedral extension of degree 2p, where p is an odd prime. Let K/F and k/F be subextensions of L/F with degrees p and 2, respectively. Then we will study relations between the p-ranks of the class groups Cl(K) and Cl(k). 1. A Short History of Reflection Theorems Results comparing the p-rank of class groups of different number fields (often based on the interplay between Kummer theory and class field theory) are traditionally called ‘reflection theorems’; the oldest such result is due to Kummer himself: let h and h denote the plus and the minus p-class number of K = Q(ζp), respectively; then Kummer observed that p | h implies p | h, and this was an important step in verifying Fermat’s Last Theorem (that is, checking the regularity of p) for exponents < 100. Kummer’s result was improved by Hecke [13] (see also Takagi [32]): Proposition 1. Let p be an odd prime, k = Q(ζp), and let Cl + p (k) and Cl − p (k) denote the plus and the minus part of Clp(k). Then rk Cl + p (k) ≤ rk Cl−p (k). Analogous inequalities hold for the eigenspaces of the class group Cl(k) under the action of the Galois group; see e.g. [15]. Scholz [30] and Reichardt [28] discovered a similar connection between the 3ranks of class groups of certain quadratic number fields: Proposition 2. Let k = Q( √ m ) with m ∈ N, and put k = Q( √ −3m ); then the 3-ranks r 3 and r − 3 of Cl(k ) and Cl(k) satisfy the inequalities r 3 ≤ r− 3 ≤ r 3 +1. Leopoldt [21] later generalized these propositions considerably and called his result the “Spiegelungssatz”. For expositions and generalizations, see Kuroda [19], Oriat [23, 26], Satgé [29], Oriat & Satgé [27], and G. Gras [10]. Damey & Payan [4] found an analog of Proposition 2 for 4-ranks of class groups of quadratic number fields: Proposition 3. Let k = Q( √ m ) be a real quadratic number field, and put k = Q( √−m ). Then the 4-ranks r 4 and r− 4 of Cl(k+) (ideal class group in the strict sense) and Cl(k) satisfy the inequalities r 4 ≤ r− 4 ≤ r 4 + 1. Other proofs were given by G. Gras [8], Halter-Koch [12], and Uehara [33]; for a generalization, see Oriat [24, 25]. In 1974, Callahan [2] discovered the following result; although it gives a connection between p-ranks of class groups of different number fields, its proof differs considerably from those of classical reflection theorems: