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
[1] A simplified version of his theorem states that if 3 divides the class number of a real quadratic field
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.
is unramified, and let E0 denote the group of primary units modulo Ep.
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.