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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 \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 \mathbb{Q} \left( \zeta_p \right)^{+} 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 \mathbb{Q} \left( \sqrt{d} \right) , then 3 also divides the class number of the imaginary quadratic field \mathbb{Q} \left( \sqrt{-3d} \right) .

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Article summary

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 \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 \mathbb{Q} \left( \zeta_p \right)^{+} 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 \mathbb{Q} \left( \sqrt{d} \right) , then 3 also divides the class number of the imaginary quadratic field \mathbb{Q} \left( \sqrt{-3d} \right) .

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  1. Summarise the article in your own words. Which single claim would you verify first, and why?
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