Definition 1.10.2  Let A be a ring and M a free A-module of finite rank and f in End_A(M). If B^tilde is an A-basis of M, we can describe f by its matrix (a_{i,j}) in B^tilde. The trace, the determinant and the characteristic polynomial of f are 

Tr(f）=  Sum_i a_{i,i} in A,

det(f) = det (a_{i,j}) in A,

chi_f(X) = det (XI_n-(a_{i,j})) in  A[X].

Proposition 1.10.4 Let L/K be a finite field extension, x in L and x_1, ..., x_n the roots (in some algebraic closure K^bar of K, counted with multiplicities) of the minimal polynomial P of x over K. Then

 Tr_{L/K}(x）= [L : K(x)] Sum_i x_i,

N_{L/K}(x) = (Prod x_i)^[L:K(x)],

chi_{x, L/K} = P^^[L:K(x)].