structural:scholar_edition
Regular representation, trace and discriminant

Recall that any field extension K / Q {\displaystyle K/\mathbb {Q} } has a unique Q {\displaystyle \mathbb {Q} } -vector space structure. Using the multiplication in K {\displaystyle K}, an element x {\displaystyle x} of the field K {\displaystyle K} over the base field Q {\displaystyle \mathbb {Q} } may be represented by n × n {\displaystyle n\times n} matrices A = A (x) = (a i j) 1 ≤ i, j ≤ n {\displaystyle A=A(x)=(a_{ij})_{1\leq i,j\leq n}} by requiring x e i = ∑ j = 1 n a i j e j, a i j ∈ Q. {\displaystyle xe_{i}=\sum _{j=1}^{n}a_{ij}e_{j},\quad a_{ij}\in \mathbb {Q}.} Here e 1, …, e n {\displaystyle e_{1},\ldots,e_{n}} is a fixed basis for K {\displaystyle K}, viewed as a Q {\displaystyle \mathbb {Q} } -vector space. The rational numbers a i j {\displaystyle a_{ij}} are uniquely determined by x {\displaystyle x} and the choice of a basis since any element of K {\displaystyle K} can be uniquely represented as a linear combination of the basis elements. This way of associating a matrix to any element of the field K {\displaystyle K} is called the regular representation. The square matrix A {\displaystyle A} represents the effect of multiplication by x {\displaystyle x} in the given basis. It follows that if the element y {\displaystyle y} of K {\displaystyle K} is represented by a matrix B {\displaystyle B}, then the product x y {\displaystyle xy} is represented by the matrix product B A {\displaystyle BA}. Invariants of matrices, such as the trace, determinant, and characteristic polynomial, depend solely on the field element x {\displaystyle x} and not on the basis. In particular, the trace of the matrix A (x) {\displaystyle A(x)} is called the trace of the field element x {\displaystyle x} and denoted Tr (x) {\displaystyle {\text{Tr}}(x)}, and the determinant is called the norm of x and denoted N (x) {\displaystyle N(x)}.

Now this can be generalized slightly by instead considering a field extension K / L {\displaystyle K/L} and giving an L {\displaystyle L} -basis for K {\displaystyle K}. Then, there is an associated matrix A K / L (x) {\displaystyle A_{K/L}(x)}, which has trace Tr K / L (x) {\displaystyle {\text{Tr}}_{K/L}(x)} and norm N K / L (x) {\displaystyle {\text{N}}_{K/L}(x)} defined as the trace and determinant of the matrix A K / L (x) {\displaystyle A_{K/L}(x)}.