Reflection theorem: Difference between revisions

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{{short description|One of several theorems linking the sizes of different ideal class groups}}

:''{{For |reflection principles in set theory, see [[reflection|Reflection principle]].''}}

{{Confusing|article|date=February 2010}}

In [[algebraic number theory]], a '''reflection theorem''' or '''Spiegelungssatz''' ([[German language|German]] for ''reflection theorem'' – see ''[[Spiegel (disambiguation)|Spiegel]]'' and ''[[Satz (disambiguation)|Satz]]'') 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>.

:''For reflection principles in set theory, see [[reflection principle]].''

In [[algebraic number theory]], a '''reflection theorem''' or '''Spiegelungssatz''' ([[German language|German]] for ''reflection theorem'' – see ''[[Spiegel]]'' and ''[[Satz (disambiguation)|Satz]]'') 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>.

==Leopoldt's Spiegelungssatz==

Both of the above results are generalized by [[Heinrich-Wolfgang Leopoldt|Leopoldt]]'s "Spiegelungssatz", which relates the [[p-rank]]s of different isotypic components of the class group of a number field considered as a [[Galois module|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 '''Q'''_''p''[''G''] and let ''A''_φ denote the corresponding direct summands of ''A''. For any φ let ''q'' = ''p''<sup>φ(1)</subsup> and let the ''G''-rank ''e''_φ be the exponent in the index

:<math> [ A_\phi : A_\phi^p ] = q^{e_\phi} . </math>

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:<math> \phi^*(g) = \omega(g) \phi(g^{-1}) . </math>

Let ''E'' be the unit group of ''K''. We say that ε is "primary" if <math>K(\sqrt[p]\epsilon)/K</math> is unramified, and let ''E''₀ denote the group of primary units modulo ''E''^''p''. Let δ_φ denote the ''G''-rank of the φ component of ''E''₀.

The Spiegelungssatz states that

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==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 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&ndash;158.</ref>. [[Kenkichi Iwasawa]] also provided an [[Iwasawa theory|Iwasawa-theoretic]] reflection theorem.

==NotesReferences==

{{reflist}}

* {{cite book | first=Helmut | last=Koch | title=Algebraic Number Theory | publisher=[[Springer-Verlag]] | year=1997 | isbn=3-540-63003-1 | zbl=0819.11044 | series=Encycl. Math. Sci. | volume=62 | edition=2nd printing of 1st | pages=147–149 }}

[[Category:Theorems in algebraic number theory]]