Let K/ℚ be a finite Galois extension with maximal order OK and Galois group Γ. For finite Γ-stable subgroups G ⊂ GLn OK) it is known , that they are generated by matrices with coefficients in OK ab, K ab the maximal abelian subextension of K over ℚ. This note gives a contribution to the corresponding question in the case of a relative Galois extension K/R, where R is a finite extension of the rationals ℚ. It turns out, that in this relative situation the answer to the corresponding question depends heavily on the arithmetic of the number field R, more precisely on the ramification behavior of primes in K/R. Due to the possibility of unramified extensions of R for certain number fields R there exist examples of Galois stable linear groups G ⊂ GLn (OK) which are not fixed elementwise by the commutator subgroup of Gal (K/R).
- Algebraic number theory
- Class numbers
- Integral representations of finite groups
- Integral representations related to algebraic numbers
ASJC Scopus subject areas
- Algebra and Number Theory
- Applied Mathematics