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Concept
Field trace
Summary
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.
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, TrL/K(α), is defined as the trace (in the linear algebra sense) of this linear transformation.
For α in L, let σ_1(α), ..., σ_n(α) be the roots (counted with multiplicity) of the minimal polynomial of α over K (in some extension field of K). Then
If L/K is separable then each root appears only once (however this does not mean the coefficient above is one; for example if α is the identity element 1 of K then the trace is [L:K ] times 1).
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.
Let be a quadratic extension of . Then a basis of is If then the matrix of is:
and so, . The minimal polynomial of α is X^2 − 2a X + (a2 − db2).
Several properties of the trace function hold for any finite extension.
The trace Tr_L/K : L → K is a K-linear map (a K-linear functional), that is
If α ∈ K 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.
Let L = GF(qn) be a finite extension of a finite field K = GF(q). 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:
For any , there are exactly elements with .
Theorem. For b ∈ L, let Fb be the map Then Fb ≠ Fc if b ≠ c. 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.
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