About: Reflection theorem

In algebraic number theory, a reflection theorem or Spiegelungssatz (German for reflection theorem – see Spiegel and Satz) is one of a collection of theorems linking the sizes of different ideal class groups (or ray class groups), or the sizes of different isotypic components of a class group. The original example is due to Ernst Eduard Kummer, who showed that the class number of the cyclotomic field , with p a prime number, will be divisible by p if the class number of the maximal real subfield is. Another example is due to Scholz. A simplified version of his theorem states that if 3 divides the class number of a real quadratic field , then 3 also divides the class number of the imaginary quadratic field .