Reduced norm
From Encyclopedia of Mathematics
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 
.
References
| [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: http://encyclopediaofmath.org/index.php?title=Reduced_norm&oldid=13357
Reduced norm. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Reduced_norm&oldid=13357