Example

(1) If A is a quadratic algebra of type $(\alpha, \beta)$ over K, then (in the notation of § 2, no. 3) $\operatorname{Tr}(e_1) = 2, \operatorname{Tr}(e_2) = \beta$,

whence $D_{A/K}(e_1, e_2) = \beta^2 + 4\alpha$.

(2) Let $A = K[X]/K[X]P$, where $P(X) = X^3 + pX + q$, so that if x is the image of X in A, 1, x, $x^2$ form a basis of A over K and $x^3 = -px - q$. It is seen immediately that $\operatorname{Tr}(1) = 3, \operatorname{Tr}(x) = 0, \operatorname{Tr}(x^2) = -2p$, taking account of the relation $x^3 = -px - q, \operatorname{Tr}(x^3) = -3q$ and $\operatorname{Tr}(x^4) = 2p^2$, whence easily $D_{A/K}(1, x, x^2) = -4p^3 - 27q^2$.

(3) Let A be a quaternion algebra of type $(\alpha, \beta, \gamma)$ over K and $(1, i, j, k)$ a basis of A of type $(\alpha, \beta, \gamma)$; taking account of § 3, no. 5, formula (30), it is easily found that $\operatorname{Tr}(1) = 4, \operatorname{Tr}(i) = 2\beta, \operatorname{Tr}(j) = \operatorname{Tr}(k) = 0$, then

(4) Let $A = M_n(K)$ and consider the canonical basis $(E_{ij})_{1 \leq i \leq n, 1 \leq j \leq n}$ of A over K (II, § 10, no. 3). It is immediate that $\operatorname{Tr}_{A/K}(E_{ij}) = 0$ if $j \neq i$ and $\operatorname{Tr}_{A/K}(E_{ii}) = n$ for all i; it follows without difficulty that the matrix $(\operatorname{Tr}(E_{ij}E_{hk}))$ of order $n^2$ is of the form $n.P$, where P is a permutation matrix, whence $D_{A/K}((E_{ij})) = \pm n^{n^2}$.