Field trace

From Wikipedia, the free encyclopedia
Jump to: navigation, search

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 (linear algebra) trace of this linear transformation.[1]

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.[2]

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 If then the matrix of is:


and so, .[1] The minimal polynomial of α is X2 - 2a X + a2 - d b2.

Properties of the trace[edit]

Several properties of the trace function hold for any finite extension.[3]

The trace TrL/K : LK 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.


Finite fields[edit]

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.[4]


In this setting we have the additional properties,[5]

Theorem.[6] For bL, let Fb be the map Then FbFc if bc. 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.[4]


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).[7] 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:


Trace form[edit]

When L/K is separable, the trace provides a duality theory via the trace form: the map from L × L to K sending (xy) 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.[8] The converse, that every Witt equivalence class with non-negative signature contains a trace form, is true for algebraic number fields K.[8]

If L/K is an inseparable extension, then the trace form is identically 0.[9]

See also[edit]


  1. ^ a b Rotman 2002, p. 940
  2. ^ Rotman 2002, p. 941
  3. ^ Roman 1995, p. 151 (1st ed.)
  4. ^ a b Lidl & Niederreiter 1997, p.54
  5. ^ Mullen & Panario 2013, p. 21
  6. ^ Lidl & Niederreiter 1997, p.56
  7. ^ Hirschfeld 1979, pp. 3-4
  8. ^ a b Lorenz (2008) p.38
  9. ^ Isaacs 1994, p. 369 as footnoted in Rotman 2002, p. 943


Further reading[edit]