Reflection Theorem - Leopoldt's Spiegelungssatz

Leopoldt's Spiegelungssatz

Both of the above results are generalized by Leopoldt's "Spiegelungssatz", which relates the p-ranks of different isotypic components of the class group of a number field considered as a module over the Galois group of a Galois extension.

Let L/K be a finite Galois extension of number fields, with group G, degree prime to p and L containing the p-th roots of unity. Let A be the p-Sylow subgroup of the class group of L. Let φ run over the irreducible characters of the group ring Q_p and let A_φ denote the corresponding direct summands of A. For any φ let q = pφ(1) and let the G-rank e_φ be the exponent in the index

Let ω be the character of G

The reflection (Spiegelung) φ* is defined by

Let E be the unit group of K. We say that ε is "primary" if is unramified, and let E₀ denote the group of primary units modulo Ep. Let δ_φ denote the G-rank of the φ component of E₀.

The Spiegelungssatz states that