Field trace

In mathematics, the field trace is a particular function defined with respect to a finite field extension L/K, which is a K-linear map from L to K.

Definition

Let K be a field and L a finite extension (and hence an algebraic extension) of K. L can be viewed as a vector space over K. Multiplication by α, an element of L,

is a K-linear transformation of this vector space into itself. The trace, Tr_L/K(α), is defined as the (linear algebra) trace of this linear transformation.

For α in L, let σ₁(α), ..., σ_n(α) be the roots (counted with multiplicity) of the minimal polynomial of α over K (in some extension field of L), then

If L/K is separable then each root appears only once and the coefficient above is one.

More particularly, if L/K is a Galois extension and α is in L, then the trace of α is the sum of all the Galois conjugates of α, i.e.

where Gal(L/K) denotes the Galois group of L/K.

Example

Let be a quadratic extension of . Then a basis of If then the matrix of is:

and so, . The minimal polynomial of α is X² - 2a X + a² - d b².