Reflection theorem
In algebraic number theory, a reflection theorem or Spiegelungssatz (German for reflection theorem – see Spiegel and Satz) is one of a collection of theorems linking the sizes of different ideal class groups (or ray class groups), or the sizes of different isotypic components of a class group. The original example is due to Ernst Eduard Kummer, who showed that the class number of the cyclotomic field , with p a prime number, will be divisible by p if the class number of the maximal real subfield  is. Another example is due to Scholz. A simplified version of his theorem states that if 3 divides the class number of a real quadratic field , then 3 also divides the class number of the imaginary quadratic field . 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 Qp[G] and let Aφ denote the corresponding direct summands of A. For any φ let q = pφ(1) and let the G-rank eφ 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 δφ denote the G-rank of the φ component of E0. The Spiegelungssatz states that

Extensions
Extensions of this Spiegelungssatz were given by Oriat and Oriat-Satge, where class groups were no longer associated with characters of the Galois group of K/k, but rather by ideals in a group ring over the Galois group of K/k. Leopoldt's Spiegelungssatz was generalized in a different direction by Kuroda, who extended it to a statement about ray class groups. This was further developed into the very general "T-S reflection theorem" of Georges Gras. Kenkichi Iwasawa also provided an Iwasawa-theoretic reflection theorem. References

 

Theorems in algebraic number theory