Reduced trace in a quaternion algebra

The reduced trace of $x$ in a quaternion algebra $B$ over a field $F$ of characteristic different from two is

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

where $\bar x$ is standard conjugation. In a presentation $B=(a,b)_F$ , one has

\operatorname{trd}(x_0+x_1i+x_2j+x_3ij)=2x_0.

Trace identities

The map is $F$ -linear and satisfies $\operatorname{trd}(xy)=\operatorname{trd}(yx)$ . Together with the reduced norm, it gives the identity

x^2-\operatorname{trd}(x)x+\operatorname{nrd}(x)=0.

This follows by expanding and using $x\bar x=\bar xx$ .

Two different traces

In $M_2(F)$ , reduced trace is ordinary matrix trace. On the four-dimensional vector space $B$ , however, the linear operator $L_x:y\mapsto xy$ has ordinary trace $2\operatorname{trd}(x)$ . Reduced trace should not be confused with the trace of that regular representation.

The kernel is the space of pure quaternions.

References

John Voight, Quaternion Algebras, author's text, §3.3, reduced trace and norm.