A neat invariant associated to a central simple algebra A/k is the reduced norm. Serre doesn't mention it because it is tangential to his aim of relating Br(k) with Galois cohomology, but it will help us give an explicit description of generalized quaternions (central simple algebras of dimension 4), and will let us give a cohomology-free proof of the triviality of the Brauer group of a finite field.

Let A be a central simple algebra of dimension n². Associated to an element x of A is the k-linear transformation M_x from A to itself, defined by M_x(y)=xy. The characteristic polynomial P_x of M_x is then an invariant of x; it is a monic polynomial of degree n² whose T^m coefficient is a homogeneous polynomial of degree n²-m in the coordinates of x. But if A is a matrix algebra then A, considered its own left regular representation, is a direct sum of n copies of the standard representation of M_n on kⁿ. Thus P_x is the n-th power of the characteristic polynomial of x when x is considered as an endomorphism of kⁿ; we'll call this polynomial the ``reduced characteristic polynomial'' of x, and denote it by R_x.

But we know that any A becomes isomorphic with M_n over an algebraic closure of k. Hence P_x is always the n-th power of some n-th degree polynomial, at least over this algebraic closure. We call this polynomial the ``reduced characteristic polynomial'' of x, and denote it by R_x. As it stands, we have defined R_x only over an algebraic closure of k; but since its n-th power is the polynomial P_x in k[T], the polynomial R_x must itself be in k[T] -- at least if A has a separable decomposition field. We shall show that this condition is always satisfied, but for the time being we have at least defined R_x as a monic polynomial of degree n in k[T] if k is perfect.

Properties of R_x: The equation R_x(x)=0 holds for all x in A. (This may be enough to prove that R_x is in k[T] even in the absence of a separable decomposition field, but I haven't tried very hard -- anyway we'll see that this is not an issue.) The T^m coefficient of R_x(x) is a homogeneous polynomial of degree n-m in the coordinates of x. In particular, its T^n-1 coefficient is a linear map from A to k, whose negative is called the reduced trace of x; when n is not a multiple of the characteristic of k, the reduced trace of x is simply the trace of M_x divided by n. Likewise, the constant coefficient of R_x(x) is (-1)ⁿ times the reduced norm of x, which is an n-th root of the determinant of M_x, and coincides with the usual determinant of x if A is M_n(k). The reduced norm is a multiplicative map from A to k, given by a polynomial of degree n in the coefficients of x. The reduced norm of x vanishes if and only if x is not invertible. In particular, if A is a division algebra then the reduced norm of x vanishes if and only if x=0.

For example: if A is the division algebra H of Hamilton quaternions over k=R, then the reduced trace of a quaternion x=a+b i+c j+d k is 2a, its reduced norm is a²+b²+c²+d², and its reduced characteristic polynomial R_x is T²-2aT+(a²+b²+c²+d²).