Reflection theorem

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For reflection principles in set theory, see reflection principle.

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 \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.[1] 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).

Leopoldt's Spiegelungssatz[edit]

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

 [ A_\phi : A_\phi^p ] = q^{e_\phi} .

Let ω be the character of G

 \zeta^g = \zeta^{\omega(g)} \text{ for } \zeta \in \mu_p .

The reflection (Spiegelung) φ* is defined by

 \phi^*(g) = \omega(g) \phi(g^{-1}) .

Let E be the unit group of K. We say that ε is "primary" if K(\sqrt[p]\epsilon)/K 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

 | e_{\phi^*} - e_\phi | \le \delta_\phi .


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.[2] Kenkichi Iwasawa also provided an Iwasawa-theoretic reflection theorem.


  1. ^ A. Scholz, Uber die Beziehung der Klassenzahlen quadratischer Korper zueinander, J. reine angew. Math., 166 (1932), 201-203.
  2. ^ Georges Gras, Class Field Theory: From Theory to Practice, Springer-Verlag, Berlin, 2004, pp. 157–158.