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 Q ( ζ p ) {displaystyle mathbb {Q} left(zeta _{p} ight)} , with p a prime number, will be divisible by p if the class number of the maximal real subfield Q ( ζ p ) + {displaystyle mathbb {Q} left(zeta _{p} ight)^{+}} 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 Q ( d ) {displaystyle mathbb {Q} left({sqrt {d}} ight)} , then 3 also divides the class number of the imaginary quadratic field Q ( − 3 d ) {displaystyle mathbb {Q} left({sqrt {-3d}} ight)} . 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 Q ( ζ p ) {displaystyle mathbb {Q} left(zeta _{p} ight)} , with p a prime number, will be divisible by p if the class number of the maximal real subfield Q ( ζ p ) + {displaystyle mathbb {Q} left(zeta _{p} ight)^{+}} 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 Q ( d ) {displaystyle mathbb {Q} left({sqrt {d}} ight)} , then 3 also divides the class number of the imaginary quadratic field Q ( − 3 d ) {displaystyle mathbb {Q} left({sqrt {-3d}} ight)} . Both of the above results are generalized by Leopoldt's 'Spiegelungssatz', which relates the p-ranks of different isotypic components of the class group of a number field considered as a module over the Galois group of a Galois extension. Let L/K be a finite Galois extension of number fields, with group G, degree prime to p and L containing the p-th roots of unity. Let A be the p-Sylow subgroup of the class group of L. Let φ run over the irreducible characters of the group ring Qp and let Aφ denote the corresponding direct summands of A. For any φ let q = pφ(1) and let the G-rank eφ be the exponent in the index Let ω be the character of G The reflection (Spiegelung) φ* is defined by Let E be the unit group of K. We say that ε is 'primary' if K ( ϵ p ) / K {displaystyle K({sqrt{epsilon }})/K} is unramified, and let E0 denote the group of primary units modulo Ep. Let δφ denote the G-rank of the φ component of E0.