Zero divisors in tensor products of division algebras
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- by Lawence J. Risman PDF
- Proc. Amer. Math. Soc. 51 (1975), 35-36 Request permission
Abstract:
Theorem. If a tensor product of a division algebra $D$ with a quaternion algebra $Q$ is not a division algebra, then either $D$ and $Q$ possess a common quadratic subfield or $D$ contains a splitting field of $Q$ not quadratic over the base field. The above theorem generalizes a recently published result of Albert’s. Theorem. If the tensor product of two division algebras over a local or a global field $K$ is not a division algebra, then they contain a common extension field of $K$.References
- A. A. Albert, Tensor products of quaternion algebras, Proc. Amer. Math. Soc. 35 (1972), 65–66. MR 297803, DOI 10.1090/S0002-9939-1972-0297803-6
- A. Adrian Albert, Structure of algebras, American Mathematical Society Colloquium Publications, Vol. XXIV, American Mathematical Society, Providence, R.I., 1961. Revised printing. MR 0123587
- Murray M. Schacher, Subfields of division rings. I, J. Algebra 9 (1968), 451–477. MR 227224, DOI 10.1016/0021-8693(68)90015-X
Additional Information
- © Copyright 1975 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 51 (1975), 35-36
- MSC: Primary 12A80; Secondary 16A40
- DOI: https://doi.org/10.1090/S0002-9939-1975-0374101-6
- MathSciNet review: 0374101