Reflection theorem

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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 ${\displaystyle \mathbb {Q} \left(\zeta _{p}\right)}$, with p a prime number, will be divisible by p if the class number of the maximal real subfield ${\displaystyle \mathbb {Q} \left(\zeta _{p}\right)^{+}}$ is. Another example is due to Scholz.^[1] A simplified version of his theorem states that if 3 divides the class number of a real quadratic field ${\displaystyle \mathbb {Q} \left({\sqrt {d}}\right)}$, then 3 also divides the class number of the imaginary quadratic field ${\displaystyle \mathbb {Q} \left({\sqrt {-3d}}\right)}$.