Reduced norm in a central simple algebra

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 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).