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,

${\displaystyle m_{\alpha }:L\to L\text{ given by }{m}_{\alpha }(x)=\alpha x}$,

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.^[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

${\displaystyle {\mathrm{Tr}}_{L/K}(\alpha )=[L:K(\alpha )]\sum _{j=1}^n{\sigma }_j(\alpha )}$.

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.,

${\displaystyle {\mathrm{Tr}}_{L/K}(\alpha )=\sum _{g\in \mathrm{Gal}(L/K)}g(\alpha ),}$

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

Example

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

${\displaystyle \left[\begin{array}{cc}a& bd\\ b& a\end{array}\right]}$,

and so, ${\displaystyle {\mathrm{Tr}}_{L/\mathbb{Q}}(\alpha )=[L:\mathbb{Q}(\alpha )]({\sigma }_1(\alpha )+{\sigma }_2(\alpha ))=1.({\sigma }_1(\alpha )+\overline{{\sigma }_1}(\alpha ))=a+b\sqrt{d}+a-b\sqrt{d}=2a}$.^[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

${\displaystyle {\mathrm{Tr}}_{L/K}(\alpha a+\beta b)=\alpha {\mathrm{Tr}}_{L/K}(a)+\beta {\mathrm{Tr}}_{L/K}(b)\text{ for all }\alpha ,\beta \in K}$.

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

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.

${\displaystyle {\mathrm{Tr}}_{M/K}={\mathrm{Tr}}_{L/K}\circ {\mathrm{Tr}}_{M/L}}$.

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.^[4]

${\displaystyle {\mathrm{Tr}}_{L/K}(\alpha )=\alpha +{\alpha }^q+\cdots +{\alpha }^{q^{n-1}}}$.

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

• ${\displaystyle {\mathrm{Tr}}_{L/K}(a^q)={\mathrm{Tr}}_{L/K}(a)\text{ for }a\in L}$
• ${\displaystyle \text{for any }\alpha \in K,\text{ we have }|\{b\in L:{\mathrm{Tr}}_{L/K}(b)=\alpha \}|=q^{n-1}}$

Theorem.^[6] For b ∈ L, let F_b be the map ${\displaystyle a\mapsto {\mathrm{Tr}}_{L/K}(ba).}$ 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 ${\displaystyle \mathrm{GF}(q)={\mathbb{F}}_q}$ 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).^[7] If b = 0 then this equation has the unique solution ${\displaystyle x=\sqrt{\frac{c}{a}}}$ in GF(q). If b ≠ 0 then the substitution y = ax/b converts the quadratic equation to the form:

${\displaystyle y^2+y+\delta =0,\text{ where }\delta =\frac{ac}{b^2}}$.

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

${\displaystyle y=s=k{\delta }^2+(k+k^2){\delta }^4+\dots +(k+k^2+\dots +k^{2^{h-2}}){\delta }^{2^{h-1}}}$.

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

${\displaystyle y=s=\delta +{\delta }^{2^2}+{\delta }^{2^4}+\dots +{\delta }^{2^{2m}}}$.

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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