Galois groups of finite abelian extensions of
Let be a finite abelian group with . Then there exist infinitely many number fields with Galois and .
This will first be proven for cyclic.
Let . By Dirichlet’s theorem on primes in arithmetic progressions, there exists a prime with . Let denote a root of unity. Let . Then is Galois with cyclic of order (http://planetmath.org/OrderGroup) . Since divides , there exists a subgroup of such that . Since is cyclic, it is abelian, and is a normal subgroup of . Let , the subfield of fixed (http://planetmath.org/FixedField) by . Then is Galois with cyclic of order . Thus, .
Let and be distinct primes with and . Then there exist subfields and of and , respectively, such that and . Note that since . Thus, . Therefore, for every prime with , there exists a distinct number field such that is Galois and . The theorem in the cyclic case follows from using the full of Dirichlet’s theorem on primes in arithmetic progressions: There exist infinitely many primes with .
|Title||Galois groups of finite abelian extensions of|
|Date of creation||2013-03-22 16:18:40|
|Last modified on||2013-03-22 16:18:40|
|Last modified by||Wkbj79 (1863)|