Zero divisors in tensor products of division algebras

Author:
Lawence J. Risman

Journal:
Proc. Amer. Math. Soc. **51** (1975), 35-36

MSC:
Primary 12A80; Secondary 16A40

MathSciNet review:
0374101

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Abstract: Theorem. *If a tensor product of a division algebra with a quaternion algebra is not a division algebra, then either and possess a common quadratic subfield or contains a splitting field of 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 is not a division algebra, then they contain a common extension field of *.

**[1]**A. A. Albert,*Tensor products of quaternion algebras*, Proc. Amer. Math. Soc.**35**(1972), 65–66. MR**0297803**, 10.1090/S0002-9939-1972-0297803-6**[2]**A. Adrian Albert,*Structure of algebras*, Revised printing. American Mathematical Society Colloquium Publications, Vol. XXIV, American Mathematical Society, Providence, R.I., 1961. MR**0123587****[3]**Murray M. Schacher,*Subfields of division rings. I*, J. Algebra**9**(1968), 451–477. MR**0227224**

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DOI:
https://doi.org/10.1090/S0002-9939-1975-0374101-6

Article copyright:
© Copyright 1975
American Mathematical Society