Reduced norm

From Encyclopedia of Mathematics
Revision as of 17:03, 7 February 2011 by (talk) (Importing text file)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search

Let be a finite-dimensional central simple algebra over . A finite extension field of is a splitting field for if as -algebras for some . Here is the -algebra of -matrices. Choose an isomorphism . The reduced norm mapping is now defined by

and the reduced trace mapping is similarly defined by

One checks that the right-hand sides of these equations are indeed in (and not just in ) and that the definitions are independent of the choices of and .

The reduced norm is multiplicative, and is invertible if and only if . The reduced trace is a homomorphism of vector spaces, and defines a non-degenerate bilinear form on .


[a1] H. Bass, "Algebraic -theory" , Benjamin (1967) pp. 152ff
[a2] A.J. Hahn, O.T. O'Meara, "The classical groups and -theory" , Springer (1979) pp. §2.2D
How to Cite This Entry:
Reduced norm. Encyclopedia of Mathematics. URL: