An Embedding Theorem for Reduced Albert Algebras Over Arbitrary Fields

Abstract

Extending two classical embedding theorems of Albert and Jacobson and Jacobson for Albert (exceptional simple Jordan) algebra over fields of characteristic not two to base fields of arbitrary characteristic, we show that any element of a reduced Albert algebra can be embedded into a reduced absolutely simple subalgebra of degree 3 and dimension 9 which may be chosen to be split if the Albert algebra was split to begin with.

Key Words:

• Absolutely simple
• Albert algebra
• Composition algebra
• Cubic Jordan algebra
• Elementary frame

2010 Mathematics Subject Classification:

• Primary 17C40
• Secondary 17A75

ACKNOWLEDGMENTS

Thanks are due to Erhard Neher for having drawn my attention to the case of nonsplit reduced Albert algebras in the embedding theorem. I am particularly grateful to Ottmar Loos, who read an earlier version of the paper with great care and made several important suggestions for improvement. Finally, my special thanks go to the referee for her or his painstaking reading of the manuscript and several useful comments which greatly contributed to further improvements of the paper.

Andrei Duma zum 70. Geburtstag gewidmet

Notes

Communicated by A. Elduque.