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This paper shows that divisible abelian torsion groups are realizable asBrauer groups of quasilocal fields. It describes the isomorphism classes ofBrauer groups of primarily quasilocal fields and solves the analogous problemconcerning the reduced components of the Brauer groups of two basic types ofHenselian valued absolutely stable fields. For a quasilocal field E and afinite separable extension R/E, we find two sufficient conditions for validityof the norm group equality $N(R/E) = N(R_{0}/E)$, where R_{0} is the maximalabelian extension of E in R. This is used for deriving information on thearising specific relations between Galois groups and norm groups of finiteGalois extensions of E.

On the Brauer groups of quasilocal fields and the norm groups of their finite Galois extensions