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claims that every element of m-torsion in Br(F) is a sum of classes of cyclic algebras of index m (t.e., each summand splits in a cyclic extension of degree m). This, of course, follows from the Merkurjev-Suslin theorem, if F contains all the m-roots of 1 -- but not otherwise.
The following argument is only nontrivial if F does not contain the (relevant) roots of unity. Every cyclic division algebra of index m splits in a certain radical extension of degree m. Indeed, cyclic algebras are the image of the map Hom(GF, Z/m) X F*/F*m --> mBr(F) (this is an easy exercise in cyclic group cohomology).
Therefore, if Merkurjev's conjecture is true, then all elements of Br(F) die in the maximal abelian extension of F, and at the same time all of them die in the maximal radical extension (obtained by adding to F all roots of all its elements).
The following argument is only nontrivial if F does not contain the (relevant) roots of unity. Every cyclic division algebra of index m splits in a certain radical extension of degree m. Indeed, cyclic algebras are the image of the map Hom(GF, Z/m) X F*/F*m --> mBr(F) (this is an easy exercise in cyclic group cohomology).
Therefore, if Merkurjev's conjecture is true, then all elements of Br(F) die in the maximal abelian extension of F, and at the same time all of them die in the maximal radical extension (obtained by adding to F all roots of all its elements).
