SupposeD is a division algebra of degreep over its centerF, which contains a primitivep-root of 1. Also supposeD has a maximal separable subfield overF whose Galois group is the semidirect product of the cyclic groupsC p C q , whereq=2, 3, 4, or 6 and is relatively prime top (In particular this is the case whenp is prime ≤7 andD has a maximal separable subfield whose Galois group is solvable.) ThenD is cyclic. The proof involves developing a theory of a wider class of algebras, which we call accessible, and proving that they are cyclic.
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Research supported in part by US-Israel Binational Science Foundation grant #92-00255. The second author is grateful for support under NSF grant DMS-9400650. Also, the authors thank the referee for several very helpful suggestions.
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Rowen, L.H., Saltman, D.J. Semidirect product division algebras. Israel J. Math. 96, 527–552 (1996). https://doi.org/10.1007/BF02937322
- Direct Summand
- Galois Group
- Prime Power
- Division Algebra
- Semidirect Product