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

[math]\displaystyle{ m_\alpha:L\to L \text{ given by } m_\alpha (x) = \alpha x }[/math],

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.^[1]

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 K), then

[math]\displaystyle{ \operatorname{Tr}_{L/K}(\alpha)=[L:K(\alpha)]\sum_{j=1}^n\sigma_j(\alpha) }[/math].

If L/K is separable then each root appears only once^[2] (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,^[1] i.e.,

[math]\displaystyle{ \operatorname{Tr}_{L/K}(\alpha)=\sum_{g\in\operatorname{Gal}(L/K)}g(\alpha), }[/math]

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

Example

Let [math]\displaystyle{ L = \mathbb{Q}(\sqrt{d}) }[/math] be a quadratic extension of [math]\displaystyle{ \mathbb{Q} }[/math]. Then a basis of [math]\displaystyle{ L/\mathbb{Q} \text{ is }\{1, \sqrt{d}\}. }[/math] If [math]\displaystyle{ \alpha = a + b\sqrt{d} }[/math] then the matrix of [math]\displaystyle{ m_{\alpha} }[/math] is:

[math]\displaystyle{ \left [ \begin{matrix} a & bd \\ b & a \end{matrix} \right ] }[/math],

and so, [math]\displaystyle{ \operatorname{Tr}_{L/\mathbb{Q}}(\alpha) = [L:\mathbb{Q}(\alpha)]\left( \sigma_1(\alpha) + \sigma_2(\alpha)\right) = 1. \left( \sigma_1(\alpha) + \overline{\sigma_1}(\alpha)\right) = a+b\sqrt{d} + a-b\sqrt{d} = 2a }[/math].^[1] 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.^[3]

The trace Tr_L/K : L → K is a K-linear map (a K-linear functional), that is

[math]\displaystyle{ \operatorname{Tr}_{L/K}(\alpha a + \beta b) = \alpha \operatorname{Tr}_{L/K}(a)+ \beta \operatorname{Tr}_{L/K}(b) \text{ for all }\alpha, \beta \in K }[/math].

If α ∈ K then [math]\displaystyle{ \operatorname{Tr}_{L/K}(\alpha) = [L:K] \alpha. }[/math]

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.

[math]\displaystyle{ \operatorname{Tr}_{M/K}=\operatorname{Tr}_{L/K}\circ\operatorname{Tr}_{M/L} }[/math].

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.^[4]

[math]\displaystyle{ \operatorname{Tr}_{L/K}(\alpha)=\alpha + \alpha^q + \cdots + \alpha^{q^{n-1}} }[/math].

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

• [math]\displaystyle{ \operatorname{Tr}_{L/K}(a^q) = \operatorname{Tr}_{L/K}(a) \text{ for } a \in L }[/math]
• [math]\displaystyle{ \text{for any }\alpha \in K, \text{ we have }|\{b \in L \colon \operatorname{Tr}_{L/K}(b) = \alpha \}| = q^{n-1} }[/math]

Theorem.^[6] For b ∈ L, let F_b be the map [math]\displaystyle{ a \mapsto \operatorname{Tr}_{L/K}(ba). }[/math] 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.^[4]

Application

A quadratic equation, ax² + bx + c = 0, with a ≠ 0, and coefficients in the finite field [math]\displaystyle{ \operatorname{GF}(q) = \mathbb{F}_q }[/math] 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).^[7] If b = 0 then this equation has the unique solution [math]\displaystyle{ x = \sqrt{\frac{c}{a}} }[/math] in GF(q). If b ≠ 0 then the substitution y = ax/b converts the quadratic equation to the form:

[math]\displaystyle{ y^2 + y + \delta = 0, \text { where } \delta = \frac{ac}{b^2} }[/math].

This equation has two solutions in GF(q) if and only if the absolute trace [math]\displaystyle{ \operatorname{Tr}_{GF(q)/GF(2)}(\delta) = 0. }[/math] 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 [math]\displaystyle{ \operatorname{Tr}_{GF(q)/GF(2)}(k) = 1. }[/math] Then a solution to the equation is given by:

[math]\displaystyle{ y = s = k \delta^2 + (k + k^2)\delta^4 + \ldots + (k + k^2 + \ldots + k^{2^{h-2}})\delta^{2^{h-1}} }[/math].

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

[math]\displaystyle{ y = s = \delta + \delta^{2^2} + \delta^{2^4} + \ldots + \delta^{2^{2m}} }[/math].

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.^[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

• Field norm
• Reduced trace

Notes

References

• Hirschfeld, J.W.P. (1979), Projective Geometries over Finite Fields, Oxford Mathematical Monographs, Oxford University Press, ISBN 0-19-853526-0, https://archive.org/details/projectivegeomet0000hirs 
• Isaacs, I.M. (1994), Algebra, A Graduate Course, Brooks/Cole Publishing 
• Lidl, Rudolf; Niederreiter, Harald (1997) [1983], Finite Fields, Encyclopedia of Mathematics and its Applications, 20 (Second ed.), Cambridge University Press, ISBN 0-521-39231-4, https://archive.org/details/finitefields0000lidl_a8r3 
• Lorenz, Falko (2008). Algebra. Volume II: Fields with Structure, Algebras and Advanced Topics. Springer. ISBN 978-0-387-72487-4. 
• Mullen, Gary L.; Panario, Daniel (2013), Handbook of Finite Fields, CRC Press, ISBN 978-1-4398-7378-6 
• Roman, Steven (2006), Field theory, Graduate Texts in Mathematics, 158 (Second ed.), Springer, Chapter 8, ISBN 978-0-387-27677-9 
• Rotman, Joseph J. (2002), Advanced Modern Algebra, Prentice Hall, ISBN 978-0-13-087868-7 

Further reading

• Conner, P.E.; Perlis, R. (1984). A Survey of Trace Forms of Algebraic Number Fields. Series in Pure Mathematics. 2. World Scientific. ISBN 9971-966-05-0. 
• Section VI.5 of Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, 211 (Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4 

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