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 onto K.

Definition

Let K be a field and L a finite 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, TrL/K, is defined as the trace of this linear transformation.
For α in L, let σ,..., σ be the roots of the minimal polynomial of α over K , then
If L/K is separable then each root appears only once.
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 denotes the Galois group of L/K.

Example">Field extension">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.

Properties of the trace

Several properties of the trace function hold for any finite extension.
The trace is a K-linear map, that is
If then
Additionally, trace behaves well in towers of fields: if M is a finite extension of L, then the trace from M to K is just the composition of the trace from M to L with the trace from L to K, i.e.

Finite fields

Let L = GF be a finite extension of a finite field K = GF. Since L/K is a Galois extension, if α is in L, then the trace of α is the sum of all the Galois conjugates of α, i.e.
In this setting we have the additional properties,
Theorem. For b ∈ L, let Fb be the map Then if. Moreover, the K-linear transformations from L to K are exactly the maps of the form Fb as b varies over the field L.
When K is the prime subfield of L, the trace is called the absolute trace and otherwise it is a relative trace.

Application

A quadratic equation,, with, and coefficients in the finite field has either 0, 1 or 2 roots in GF. If the characteristic of GF is odd, the discriminant, indicates the number of roots in GF and the classical quadratic formula gives the roots. However, when GF has even characteristic, these formulas are no longer applicable.
Consider the quadratic equation with coefficients in the finite field GF. If b = 0 then this equation has the unique solution in GF. If then the substitution converts the quadratic equation to the form:
This equation has two solutions in GF if and only if the absolute trace In this case, if is one of the solutions, then is the other. Let k be any element of GF with Then a solution to the equation is given by:
When h = 2m + 1, a solution is given by the simpler expression:

Trace form

When L/K is separable, the trace provides a duality theory via the trace form: the map from to K sending to Tr is a nondegenerate, symmetric, bilinear form called the trace form. An example of where this is used is in algebraic number theory in the theory of the different ideal.
The trace form for a finite degree field extension L/K has non-negative signature for any field ordering of K. The converse, that every Witt equivalence class with non-negative signature contains a trace form, is true for algebraic number fields K.
If L/K is an inseparable extension, then the trace form is identically 0.