ON K₂ OF DIVISION ALGEBRAS

ABSTRACT

In this paper, it is proved that if F is a global field, then for any integer n > 3, there is an extension field E over F of degree n such that K ₂ E is not generated by the Steinberg symbols {a, b} with a ∈ F*, b, ∈ E*. If however, F is a number field and D is a finite-dimensional central division F-algebra with square free index, then K ₂ D is always generated by the Steinberg symbols {a, b} with a ∈ F*, b ∈ D*. Finally, the tame kernels of central division algebras over F are expressed explicitly.

Key Words:

• Central division algebra
• K ₂ group
• Reduced norm

1991 MR Subject Classification:

• 19C99
• 19F27
• 11S45

ACKNOWLEDGMENTS

The authors are deeply grateful to Professor Fei Xu for valuable discussion. Proposition 2.2 was suggested by Professor Fei Xu. The first author would like to thank the Morningside Center of Mathematics for hospitality. Supported by National Distinguished Young Science Foundation of China Grant and the 973 Grant.

Notes

^#Communicated by C. Pedrini.