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add disambiguating hat-note : For reflection principles in set theory, see reflection principle. |
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:''For reflection principles in set theory, see [[reflection principle]].''
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>.
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