Reflection theorem

Reflection theorem For reflection principles in set theory, see Reflection principle. This article may be confusing or unclear to readers. Please help clarify the article. There might be a discussion about this on the talk page. (February 2010) (Learn how and when to remove this template message) 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 {displaystyle 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 {displaystyle 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 {displaystyle mathbb {Q} left({sqrt {d}}right)} , then 3 also divides the class number of the imaginary quadratic field {displaystyle mathbb {Q} left({sqrt {-3d}}right)} .

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 {displaystyle [A_{phi }:A_{phi }^{p}]=q^{e_{phi }}.} Let ω be the character of G {displaystyle zeta ^{g}=zeta ^{omega (g)}{text{ for }}zeta in mu _{p}.} The reflection (Spiegelung) φ* is defined by {displaystyle phi ^{*}(g)=omega (g)phi (g^{-1}).} Let E be the unit group of K. We say that ε is "primary" if {displaystyle 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 {displaystyle |e_{phi ^{*}}-e_{phi }|leq delta _{phi }.} 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.[2] Kenkichi Iwasawa also provided an Iwasawa-theoretic reflection theorem.

References ^ A. Scholz, Uber die Beziehung der Klassenzahlen quadratischer Korper zueinander, J. reine angew. Math., 166 (1932), 201-203. ^ Georges Gras, Class Field Theory: From Theory to Practice, Springer-Verlag, Berlin, 2004, pp. 157–158. Koch, Helmut (1997). Algebraic Number Theory. Encycl. Math. Sci. Vol. 62 (2nd printing of 1st ed.). Springer-Verlag. pp. 147–149. ISBN 3-540-63003-1. Zbl 0819.11044. Categories: Theorems in algebraic number theory

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