Reduced norm

2020 Mathematics Subject Classification: Primary: 16H05 [MSN][ZBL]

Let $A$ be a finite-dimensional 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

[Ba]	H. Bass, "Algebraic K-theory", Benjamin (1967) pp. 152ff MR0279159 Zbl 0226.13006
[HaOM]	A.J. Hahn, O.T. O'Meara, "The classical groups and $K$-theory", Springer (1979) pp. §2.2D MR1007302 Zbl 0683.20033