For α in L, let σ1(α), ..., σn(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.
where Gal(L/K) denotes the Galois group of L/K.
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.
Properties of the trace
Several properties of the trace function hold for any finite extension.
The trace TrL/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.
In this setting we have the additional properties,
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.
A quadratic equation, and coefficients in the finite field 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 However, when GF(q) has even characteristic (i.e., q = 2h for some positive integer h), these formulas are no longer applicable.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(2h). If b = 0 then this equation has the unique solution in GF(q). If b ≠ 0 then the substitution y = ax/b converts the quadratic equation to the form:
This equation has two solutions in GF(q) if and only if the absolute trace 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 Then a solution to the equation is given by:
When h = 2m + 1, a solution is given by the simpler expression:
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 TrL/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.
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