Reflection theorem: Difference between revisions

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==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 ''K'', considered as a [[Galois module|module]] over the [[Galois group]] of a Galois extension ''K''/''k''.

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)</sub> and let the ''G''-rank ''e''_φ be the exponent in the index

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

Let ω be the character of ''G''

:<math> \zeta^g = \zeta^{\omega(g)} \text{~for~} \zeta \in \mu_p . </math>

The reflection (''Spiegelung'') φ^* is defined by

:<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

:<math> | e_{\phi^*} - e_\phi | \le \delta_\phi . </math>

==Extensions==