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

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

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 product of quaternion algebras*, Proc. Amer. Math. Soc.**35**(1972), 65-66. MR**45**#6855. MR**0297803 (45:6855)****[2]**-,*Structure of algebras*, rev. printing, Amer. Math. Soc. Colloq. Publ., vol. 24, Amer. Math. Soc., Providence, R. I., 1961. MR**23**#A912. MR**0123587 (23:A912)****[3]**Murray Schacher,*Subfields of division rings*. I, J. Algebra**9**(1968), 451-477. MR**37**#2809. MR**0227224 (37:2809)**

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

Article copyright:
© Copyright 1975
American Mathematical Society