Definition 3

Let A be a K-algebra admitting a finite basis of n elements. The discriminant of a sequence $(x_1, \ldots, x_n)$ of n elements of A, with respect to K, denoted by $D_{A/\mathbf{K}}(x_1, \ldots, x_n)$, is the discriminant of the square matrix

Consider first a basis $(e_i)_{1 \leq i \leq n}$ of A over K and write Then $\operatorname{Tr}_{A/\mathbf{K}}(e_i) = \sum_{s=1}^n c_{iss}$, whence $\operatorname{Tr}_{A/\mathbf{K}}(e_i e_j) = \sum_{k,s} c_{ij k} c_{kss}$ and therefore Now let $(x_i)_{1 \leq i \leq n}, (x'_i)_{1 \leq i \leq n}$ be two sequences of n elements of A and suppose that there exists a square matrix of order n, $M = (m_{ij})$, with coefficients in K, such that $x_i = \sum_{j=1}^n m_{ij} x'_j$ for $1 \leq i \leq n$. We write Then $\operatorname{Tr}_{A/\mathbf{K}}(x_i x_j) = \sum_{p,q} m_{ip} m_{jq} \operatorname{Tr}_{A/\mathbf{K}}(x'_p x'_q)$, whence $T = M \cdot T' \cdot {}^t M$; the rule of multiplication of determinants therefore gives whence finally

The above formula shows in particular that the discriminants of two bases of A over K differ by the square of an invertible element of K and therefore generate the same (principal) ideal of K. This ideal $\Delta_{A/K}$ is called the discriminant ideal of A over K; by formula (32) the discriminant of every sequence of n elements of A which differ only in the order of the terms have the same discriminant, for the determinant of a permutation matrix is equal to $\pm 1$.