Reflection theorem: Difference between revisions

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A '''reflection theorem''' (or '''Spiegelungssatz''') in [[algebraic number theory]] is one of a collection of theorems linking the sizes of different [[ideal class group]]s (or [[ray class group]]s), or the sizes of different [[isotypic component]]s of a class group. The original example is due to [[Ernst Kummer|Ernst Eduard Kummer]], who showed that the class number of the [[cyclotomic field]] <math>\mathbb{Q} \left( \zeta_p \right)</math>, with ''p'' a prime number, will be divisible by ''p'' if the class number of the maximal real subfield <math>\mathbb{Q} \left( \zeta_p \right)^{+}</math> is. Another example is due to Scholz<ref>A. Scholz, Uber die Beziehung der Klassenzahlen quadratischer Korper zueinander, ''J. reine angew. Math.'', '''166''' (1932), 201-203.</ref>. A simplified version of his theorem states that if 3 divides the class number of a [[real quadratic field]] <math>\mathbb{Q} \left( \sqrt{d} \right)</math>, then 3 also divides the class number of the [[imaginary quadratic field]] <math>\mathbb{Q} \left( \sqrt{-3d} \right)</math>.

Both of the above results are generalized by [[Leopoldt]]'s "SpiegelugssatzSpiegelungssatz", which relates the [[p-rank]]s of different isotypic components of the class group of a number field <math>K</math>, considered as a [[Galois module|module]] over the [[Galois group]] of a Galois extension <math>K/k</math>. 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 <math>K/k</math>, but rather by ideals in a [[group ring]] over the Galois group of <math>K/k</math>. Leopoldt's Spiegelungssatz was generalized in a different direction by Kuroda, who extended it to a statement about [[ray class group]]s. This was further developed into the very general "''T''-''S'' reflection theorem" of [[Georges Gras]]<ref>Georges Gras, ''Class Field Theory: From Theory to Practice'', Springer-Verlag, Berlin, 2004, pp. 157-158.</ref>. [[Kenkichi Iwasawa]] also provided an [[Iwasawa theory|Iwasawa-theoretic]] reflection theorem.

==Notes==