Multiplicative groups of fields modulo products of subfields

Let E i , 1 ≤ i ≤ r, be intermediate fields of the finite separable field extension K k . We study the quotient K ∗ E 1 ∗ … E r ∗ . 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 sui...

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Bibliographic Details
Published in:Journal of pure and applied algebra 1996-02, Vol.106 (3), p.233-262
Main Authors: Colliot-Thélène, J.-L., Guralnick, R.M., Wiegand, R.
Format: Article
Language:English
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Summary:Let E i , 1 ≤ i ≤ r, be intermediate fields of the finite separable field extension K k . We study the quotient K ∗ E 1 ∗ … E r ∗ . 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 K ∗ E 1 ∗ … E r ∗ 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.
ISSN:0022-4049
1873-1376
DOI:10.1016/0022-4049(94)00128-6