View source for Reduced norm


{{MSC|16H05}}
{{TEX|done}}

Let $A$ be a finite-dimensional
[[Central simple algebra|central simple algebra]] over $k$. A finite extension field $K$ of $k$ is a splitting field for $A$ if $\def\M{\textrm{M}} A\otimes_k K\simeq \M_m(K)$ as $K$-algebras for some $m$. Here $\M_m(K)$ is the $K$-algebra of $(m\times m)$-matrices. Choose an isomorphism $\def\phi{\varphi} \phi:A\otimes_k K \to \M_m(K)$. The reduced norm mapping $\def\Nrd{\textrm{Nrd}} \Nrd_{A/k}:A\to k$ is now defined by

$$\Nrd_{A/k}(a) = \det(\phi(a\otimes 1)),$$
and the reduced trace mapping $\def\Trd{\textrm{Trd}} \Trd_{A/k}(a)$ is similarly defined by

$$\Trd_{A/k}(a) = \textrm{trace}(\phi(a\otimes1)).$$
One checks that the right-hand sides of these equations are indeed in $k$ (and not just in $K$) and that the definitions are independent of the choices of $\phi$ and $K$.

The reduced norm is multiplicative, and $a\in A$ is invertible if and only if $\Nrd_{A/k}(a) \ne 0$. The reduced trace is a homomorphism of $k$-vector spaces, and $(x,y)\mapsto \Trd_{A/k}(xy)$ defines a non-degenerate bilinear form on $A$.

====References====
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|valign="top"|{{Ref|Ba}}||valign="top"|  H. Bass,   "Algebraic K-theory", Benjamin  (1967)  pp. 152ff  {{MR|0279159}}  {{ZBL|0226.13006}}    
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|valign="top"|{{Ref|HaOM}}||valign="top"|  A.J. Hahn,   O.T. O'Meara,   "The classical groups and $K$-theory", Springer  (1979)  pp. §2.2D    {{MR|1007302}}   {{ZBL|0683.20033}}  
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