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 n^{2}.
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 n^{2}
whose T^{m} coefficient is a homogeneous polynomial
of degree n^{2}-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 k^{n}.
Thus P_{x} is the n-th power of the characteristic polynomial
of x when x is considered as an endomorphism of k^{n};
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
a^{2}+b^{2}+c^{2}+d^{2},
and its reduced characteristic polynomial R_{x} is
T^{2}-2aT+(a^{2}+b^{2}+c^{2}+d^{2}).