Field trace - Alchetron, The Free Social Encyclopedia

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

m α : L → L given by m α ( x ) = α x ,

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

Tr L / K ⁡ ( α ) = [ L : K ( α ) ] ∑ j = 1 n σ j ( α ) .

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.

Tr L / K ⁡ ( α ) = ∑ g ∈ Gal ⁡ ( L / K ) g ( α ) ,

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

Example

Let L = Q ( d ) be a quadratic extension of Q . Then a basis of L / Q is { 1 , d } . If α = a + b d then the matrix of m α is:

[ a b d b a ] ,

and so, Tr L / Q ⁡ ( α ) = 2 a . The minimal polynomial of α is X² - 2a X + a² - d b².

Properties of the trace

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

Tr L / K ⁡ ( α a + β b ) = α Tr L / K ⁡ ( a ) + β Tr L / K ⁡ ( b ) for all α , β ∈ K .

If α ∈ K then Tr L / K ⁡ ( α ) = [ L : K ] α .

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.

Tr M / K = Tr L / K ∘ Tr M / L .

Finite fields

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.

Tr L / K ⁡ ( α ) = α + α q + ⋯ + α q n − 1 .

In this setting we have the additional properties,

Tr L / K ⁡ ( a q ) = Tr L / K ⁡ ( a ) for a ∈ L

for any α ∈ K , we have | { b ∈ L : Tr L / K ⁡ ( b ) = α } | = q n − 1

Theorem. For b ∈ L, let F_b be the map a ↦ Tr L / K ⁡ ( b a ) . Then F_b ≠ F_c if b ≠ c. Moreover the K-linear transformations from L to K are exactly the maps of the form F_b 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, a x 2 + b x + c = 0 , with a ≠ 0 , and coefficients in the finite field GF ⁡ ( q ) = F q has either 0, 1 or 2 roots in GF(q) (and two roots, counted with multiplicity, in the quadratic extension GF(q²)). If the characteristic of GF(q) is odd, the discriminant, Δ = b² - 4ac indicates the number of roots in GF(q) and the classical quadratic formula gives the roots. However, when GF(q) has even characteristic (i.e., q = 2^h for some positive integer h), these formulas are no longer applicable.

Consider the quadratic equation ax² + bx + c = 0 with coefficients in the finite field GF(2^h). If b = 0 then this equation has the unique solution x = c a in GF(q). If b ≠ 0 then the substitution y = ax/b converts the quadratic equation to the form:

y 2 + y + δ = 0 , where δ = a c b 2 .

This equation has two solutions in GF(q) if and only if the absolute trace Tr G F ( q ) / G F ( 2 ) ⁡ ( δ ) = 0. In this case, if y = s is one of the solutions, then y = s + 1 is the other. Let k be any element of GF(q) with Tr G F ( q ) / G F ( 2 ) ⁡ ( k ) = 1. Then a solution to the equation is given by:

y = s = k δ 2 + ( k + k 2 ) δ 4 + … + ( k + k 2 + … + k 2 h − 2 ) δ 2 h − 1 .

When h = 2m + 1, a solution is given by the simpler expression:

y = s = δ + δ 2 2 + δ 2 4 + … + δ 2 2 m .

Trace form

When L/K is separable, the trace provides a duality theory via the trace form: the map from L × L to K sending (x, y) to Tr_L/K(xy) 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.

References

Field trace Wikipedia

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