Reflection theorem

For reflection principles in set theory, see reflection principle.

A reflection theorem (or Spiegelungssatz) in algebraic number theory 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 $\mathbb {Q} \left(\zeta _{p}\right)$, with p a prime number, will be divisible by p if the class number of the maximal real subfield $\mathbb {Q} \left(\zeta _{p}\right)^{+}$ 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 $\mathbb {Q} \left({\sqrt {d}}\right)$, then 3 also divides the class number of the imaginary quadratic field $\mathbb {Q} \left({\sqrt {-3d}}\right)$.

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 $K$, considered as a module over the Galois group of a Galois extension $K/k$. Extensions of his 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.