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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. 157–158.</ref>. [[Kenkichi Iwasawa]] also provided an [[Iwasawa theory|Iwasawa-theoretic]] reflection theorem.
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{{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]]
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