Multiplicative groups of fields modulo products of subfields Original Research Article

Let Ei, 1 ≤ i ≤ r, be intermediate fields of the finite separable field extension image. We study the quotient image. We show that there is a dichotomy between the cases r ≤ 2 and r > 2. If r ≤ 2, then the n-torsion subgroup of that quotient is finite for all n > 0, and under suitable hypotheses the entire torsion subgroup is finite. For r > 2, examples are given to show that the group image may be trivial, finite and nontrivial, infinite torsion or may have infinite torsionfree rank. The case r = 2 had been considered earlier in connection with the study of Picard groups of certain singular curves. In the present paper, we study the problem in the more general context of a finite group acting on a module, and then use Galois cohomology.