Reflection Theorem

In algebraic number theory, a reflection theorem, known in German as a Spiegelungssatz, is any of a family of results relating the sizes of ideal class groups and ray class groups of related number fields. Ernst Eduard Kummer gave the first example, showing that a prime p divides the class number of the cyclotomic field generated by a p-th root of unity whenever it divides the class number of that field's maximal real subfield, and the Scholz reflection principle later showed that if 3 divides the class number of a real quadratic field then 3 also divides the class number of the related imaginary quadratic field. Heinrich-Wolfgang Leopoldt generalized these results into the Spiegelungssatz that bears his name, applicable to arbitrary Galois extensions of number fields, bounding the difference between the p-ranks of reflected characters of a class group. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/