Every Galois lattice is the character lattice of a torus #
A continuous finite free integral representation of the absolute Galois group factors through a finite Galois extension. Descending the group algebra along that extension produces a torus. Its geometric characters recover the given representation, including the absolute-Galois action. Thus the character-lattice functor is essentially surjective over every field, including imperfect fields.
The finite extension is GaloisLatticeCat.separableActionField; the torus and its
character comparison use the invariant group-algebra constructions of GaloisDescent.
References #
- J. S. Milne, Algebraic Groups (2017), Theorem 12.23 and Corollary 12.24.
Every continuous integral Galois lattice occurs as the geometric character lattice of a torus. This is the essential-surjectivity half of the classification of tori.