Reduced norm in a quaternion algebra

The reduced norm of $x$ in a quaternion algebra $B=(a,b)_F$ is

\operatorname{nrd}(x)=x\bar x\in F,

using standard conjugation. For $x=x_0+x_1i+x_2j+x_3ij$ , it is

\operatorname{nrd}(x)=x_0^2-a x_1^2-b x_2^2+ab x_3^2.

Multiplication and inversion

Because $y\bar y$ is central, $(xy)\overline{xy}=x(y\bar y)\bar x$ , proving multiplicativity. An element is invertible exactly when its reduced norm is nonzero, with $x^{-1}=\bar x/\operatorname{nrd}(x)$ .

Split case

For $B=M_2(F)$ , the reduced norm is the determinant. Over a general field it is not a positive metric norm; it can vanish on a nonzero element of a split algebra.

Trace-zero elements

The companion reduced trace is $\operatorname{trd}(x)=x+\bar x$ . A pure quaternion has trace zero and satisfies $x^2=-\operatorname{nrd}(x)$ .