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
Let A be a central simple algebra of dimension n2. Associated to an element x of A is the k-linear transformation Mx from A to itself, defined by Mx(y)=xy. The characteristic polynomial Px of Mx is then an invariant of x; it is a monic polynomial of degree n2 whose Tm 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 Mn on kn. Thus Px 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 Rx.
But we know that any A becomes isomorphic with Mn over an algebraic closure of k. Hence Px 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 Rx. As it stands, we have defined Rx only over an algebraic closure of k; but since its n-th power is the polynomial Px in k[T], the polynomial Rx 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 Rx as a monic polynomial of degree n in k[T] if k is perfect.
Properties of Rx: The equation Rx(x)=0 holds for all x in A. (This may be enough to prove that Rx 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 Tm coefficient of Rx(x) is a homogeneous polynomial of degree n-m in the coordinates of x. In particular, its Tn-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 Mx divided by n. Likewise, the constant coefficient of Rx(x) is (-1)n times the reduced norm of x, which is an n-th root of the determinant of Mx, and coincides with the usual determinant of x if A is Mn(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 Rx is T2-2aT+(a2+b2+c2+d2).