Algebraic Number Field - Regular Representation, Trace and Determinant

Regular Representation, Trace and Determinant

Using the multiplication in F, the elements of the field F may be represented by n-by-n matrices

A = A(x)=(a_ij)_{1 ≤ i, j ≤ n},

by requiring

Here e₁, ..., e_n is a fixed basis for F, viewed as a Q-vector space. The rational numbers a_ij are uniquely determined by x and the choice of a basis since any element of F can be uniquely represented as a linear combination of the basis elements. This way of associating a matrix to any element of the field F is called the regular representation. The square matrix A represents the effect of multiplication by x in the given basis. It follows that if the element y of F is represented by a matrix B, then the product xy is represented by the matrix product AB. Invariants of matrices, such as the trace, determinant, and characteristic polynomial, depend solely on the field element x and not on the basis. In particular, the trace of the matrix A(x) is called the trace of the field element x and denoted Tr(x), and the determinant is called the norm of x and denoted N(x).

By definition, standard properties of traces and determinants of matrices carry over to Tr and N: Tr(x) is a linear function of x, as expressed by Tr(x + y) = Tr(x) + Tr(y), Tr(λx) = λ Tr(x), and the norm is a multiplicative homogeneous function of degree n: N(xy) = N(x) N(y), N(λx) = λn N(x). Here λ is a rational number, and x, y are any two elements of F.

The trace form derives is a bilinear form defined by means of the trace, as Tr(x y). The integral trace form, an integer-valued symmetric matrix is defined as t_ij = Tr(b_ib_j), where b₁, ..., b_n is an integral basis for F. The discriminant of F is defined as det(t). It is an integer, and is an invariant property of the field F, not depending on the choice of integral basis.

The matrix associated to an element x of F can also be used to give other, equivalent descriptions of algebraic integers. An element x of F is an algebraic integer if and only if the characteristic polynomial p_A of the matrix A associated to x is a monic polynomial with integer coefficients. Suppose that the matrix A that represents an element x has integer entries in some basis e. By the Cayley–Hamilton theorem, p_A(A) = 0, and it follows that p_A(x) = 0, so that x is an algebraic integer. Conversely, if x is an element of F which is a root of a monic polynomial with integer coefficients then the same property holds for the corresponding matrix A. In this case it can be proven that A is an integer matrix in a suitable basis of F. Note that the property of being an algebraic integer is defined in a way that is independent of a choice of a basis in F.